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INTELLECTUAL 


ABIT  II  ME  TIC, 


INDUCTIVE    METHOD 


INSTRUCTION. 


BY    WARil^N    COLBURN,    A.    M. 


IIIIBOTYPED  AT  THE  BOSTON  TYPE  AND  STEREOTYPE  For 


BOSTON: 

HILLIALD,  GRAY,  «fc  CO"JF 

ism , 


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* 


'Tiss   Sue  Dunbar 


Educ •   Library 


I 


*  ,  C   :* 


COLBURN'S    FIRST    LESSONS. 


INTELLECTUAL 


ARITHMETIC, 


UPON   THE 


INDUCTIVE    METHOD 


INSTRUCTION. 


BY  WARREN   COLBURN,   A.M. 


STEREOTYPED  AT  THE  BOSTON  TYPE  AND  STEREOTYPE  FOUNDRY. 


BOSTON: 

• 

PUBLISHED    BY 

HILLIARD,   GRAY,    AND    CO* 
1836. 


DISTRICT  OF  MASSACHUSETTS,  to  trft  .« 

District  Clerk's  Office. 

BB  IT  REMEMBERED,  That  on  the  twenty-third  day  of  March,  A.  D.  J82fi, 
in  the  fiftieth  year  of  the  Independence  of  the  United  States  of  America, 
CUMMIXOS,  MILLIARD,  AND  COMPANY,  of  the  said  district,  have  deposited 
in  this  office  the  title  of  a  book,  the  right  whereof  they  claim  as  proprie- 
tors, in  the  words  following,  to  wit: 

"  Intellectual  Arithmetic,  upon  the  Inductive  Method  of  Instruction.  By 
Warren  Colburn,  A.  M." 

In  conformity  to  the  act  of  the  Congress  of  the  United  States,  entitled, 
u  An  Act  for  the  encouragement  of  learning,  by  securing  the  copies  of  maps, 
charts,  and  books,  to  the  authors  and  proprietors  of  such  copies  during  the 
times  therein  mentioned;"  and  also  to  an  act,  entitled,  "  An  Act  supple- 
mentary to  an  act,  entitled, '  An  Act  for  the  encouragement  of  learning,  by 
securing  the  copies  of  maps,  charts,  and  books,  to  the  authors  and  proprie- 
tors of  such  copies  during  the  times  therein  mentioned,' and  extending  tho 
benefits  thereof  to  the  arts  of  designing,  engraving,  and  etching  historical 
and  other  prints."  JOHN  W.  DAVIS, 

Clerk  of  the  District  tf  Massachusetts. 


RECOMMENDATIONS. 

Sir,  BOSTON,  15  November,  1821. 

I  have  made  use  of  the  Arithmetic  and  Tables,  which  you  some- 
time since  prepared,  on  the  system  of  Pestalozzi ;  and  have  been  much 
f  ratified  with  the  improved  edition  of  it;  which  you  have  shown  me. 
am  satisfied,  from  experiment,  that  it  is  the  most  effectual  and  in- 
teresting mode  of  teaching  the  science  of  numbers  with  which  I  am 
acquainted.  Respectfully, 

Your  obedient  servant, 

HENRY  COLMAN. 
Mr.  WARREN  COLBURN. 


Having  been  made  acquainted  with  Mr.  Colburn's  treatise  on  Arith- 
metic, ajid  having  attended  an  examination  of  his  scholars,  who  had  been 
taught  according  to  this  system,  I  am  well  satisfied  that  it  is  the  most 
easy,  simple,  and  natural  way  of  introducing  young  persons  to  the  first 
principles  in  the  science  of  numbers.  The  method  here  proposed  is 
the  fruit  of  much  study  and  reflection.  The  author  has  had  considera- 
ble experience  as  a  teacher,  added  to  a  strong  interest  in  the  subject, 
and  a  thorough  knowledge  not  only  of  this  but  of  many  of  the  higher 
branches  of  mathematics.  This  little  work  is  therefore  earnestly 
recommended  to  the  notice  of  those  who  are  employed  in  this  branch 
of  early  instruction,  with  the  belief  that  it  only  requires  a  fair  trial  in 
order  to  be  fully  approved  and  adopted. 

J.  FARRAR, 
Prof.  Math.  Harvard  University. 

CAMBRIDGE,  Nov.  1C,  182L 


Ex 


PREFACE.      /,, 


As  soon  as  a  child  begins  to  use  his  senses,  nature  contin- 
ually presents  to  his  eyes  a  variety  of  objects  ;  and  one  of  the 
first  properties  which  he  discovers,  is  the  relation  of  number. 
He  intuitively  fixes  upon  unity  as  a  measure,  and  from  this 
he  forms  the  idea  of  more  and  less;  which  is  the  idea  of 
quantity. 

The  names  of  a  few  of  the  first  numbers  are  usually  learned 
very  early  ;  and  children  frequently  learn  to  count  as  far  as  a 
hundred  before  they  learn  their  letters. 

As  soon  as  children  have  the  idea  of  more  and  less,  and  the 
names  of  a  few  of  the  first  numbers,  they  are  able  to  make  small 
calculations.  And  this  we  see  them  do  every  day  about  their 
playthings,  and  about  the  little  affairs  which  they  are  called 
upon  to  attend  to.  The  idea  of  more  and  less  implies  addition  ; 
hence  they  will  often  perform  these  operations  without  any 
previous  instruction.  If.  for  example,  one  child  has  three 
apples,  and  another  five,  they  will  readily  tell  how  many  they 
both  have  ;  and  how  many  one  has  more  than  the  other.  If  a 
child  be  requested  to  bring  three  apples  for  each  person  in 
the  room,  he  will  calculate  very  readily  how  many  to  bring, 
if  the  number  does  not  exceed  those  he  has  learnt.  Again, 
if  a  child  be  requested  to  divide  a  number  of  apples  among 
a  certain  number  of  persons,  he  will  contrive  a  way  to  do  it, 
and  will  tell  how  many  each  must  have.  The  method  which 
children  take  to  do  these  things,  though  always  correct,  is  not  . 
always  the  most  expeditious. 

The  fondness  which  children  usually  manifest  for  these 
exercises,  and  the  facility  with  which  they  perform  them, 
seem  to  indicate  that  the  science  of  numbers,  to  a  certain 
extent,  should  be  among  the  first  lessons  taught  to  them.* 

To  succeed  in  this,  however,  it  is  necessary  rather  to  fur- 
nish occasions  for  them  to  exercise  their  own  skill  in  per- 
forming examples,  than  to  give  them  rules.  They  should  be 
allowed  to  pursue  their  own  method  first,  and  then  they 
should  be  ma*de  to  observe  and  explain  it;  and,  if  it  was  not 

*  See,  on  this  subject,  two  essays,  entitled  Juvenile  Studies,  in  the 
Prize  Book  of  the  Latin  School,  Nos.  I.  and  IP.,  published  by  Cum- 
mings  &  Hilliard,  1820  and  1821. 

&291992! 


4  PREFACE. 

the  beat,  some  improvement  should  be  suggested.  By  follow- 
ing this  mode,  and  making  the  examples  gradually  increase 
in  difficulty,  experience  proves,  that,  at  an  early  age,  children 
may  be  taught  a  great  variety  of  the  most  useful  combinations 
of  numbers. 

Few  exercises  strengthen  and  mature  the  mind  so  much  as 
arithmetical  calculations,  if  the  examples  are  made  sufficiently 
simple  to  be  understood  by  the  pupil;  because  a  regular, 
though  simple  process  of  reasoning,  is  requisite  to  perform 
them,  and  the  results  are  attended  with  certainty. 

The  idea  of  number  is  first  acquired  by  observing  sensible 
objects.  Having  observed  that  this  quality  is  common  to  all 
things  with  which  we  are  acquainted,  we  obtain  an  abstract 
idea  of  number.  We  first  make  calculations  about  sensible 
objects ;  and  we  soon  observe,  that  the  same  calculations 
will  apply  to  things  very  dissimilar ;  and,  finally,  that  they 
may  be  made  without  reference  to  any  particular  things. 
Hence  from  particulars  we  establish  general  principles,  which 
serve  as  the  basis  of  our  reasonings,  and  enable  us  to  proceed, 
step  by  step,  from  the  most  simple  to  the  more  complex  opera- 
tions. It  appears,  therefore,  that  mathematical  reasoning 
proceeds  as  much  upon  the  principle  of  analytic  induction,  as 
that  of  any  other  science. 

Examples  of  any  kind  upon  abstract  numbers,  are  of  very 
little  use,  until  the  learner  has  discovered  the  principle  from 
practical  examples.  They  are  more  difficult  in  themselves, 
for  the  learner  does  not  see  their  use ;  and  therefore  does 
not  so  readily  understand  the  question.  But  questions  of  a 
practical  kind,  if  judiciously  chosen,  show  at  once  what  the 
combination  is,  and  what  is  to  be  effected  by  it.  Plence  the 
pupil  will  much  more  readily  discover  the  means  by  which 
the  result  is  to  be  obtained.  The  mind  is  also  greatly  assisted 
%in  the  operations  by  reference  to  sensible  objects.  When  the 
pupil  learns  a  new  combination  by  means  of  abstract  examples, 
it  very  seldom  happens  that  he  understands  practical  examples 
more  easily  for  it.  because  he  does  not  discover  the  connection 
until  he  has  performed  several  practical  examples,  and  begins 
to  generalize  them. 

After  the  pupil  comprehends  an  operation,  abstract  examples 
are  useful  to  exercise  him,  and  make  him  familiar  with  it. 
And  they  serve  better  to  fix  the  principle,  because  they  teach 
the  learner  to  generalize. 

From  the  above  observations,  and  from  his  own*  experience, 
the  author  bos  been  induced  to  publish  this  treatise  ;  in  which 
he  has  pursued  the  following  plan,  which  seemed  to  him  the 
most  agreeable  to  the  natural  progress  of  the  mind. 


PREFACE. 


GENERAL  VIEW  OF  THE  PLAN. 

EVERY  combination  commences  with  practical  examples. 
Care  has  been  taken  to  select  such  as  will  aptly  illustrate  the 
combination,  and  assist  the  imagination  of  the  pupil  in  per- 
forming it.  In  most  instances,  immediately  after  the  prac- 
tical, abstract  examples  are  placed,  containing  the  same 
numbers  and  the  same  operations,  that  the  pupil  may  the 
more  easily  observe  the  connection.  The  instructer  should 
be  careful  to  make  the  pupil  observe  the  connection.  After 
these,  are  a  few  abstract  examples,  and  then  practical  questions 
again. 

The  numbers  are  small,  and  the  questions  so  simple,  that 
almost  any  child  of  five  or  six  years  old  is  capable  of  under- 
standing more  than  half  the  book,  and  those  of  seven  or  eight 
years  old  can  understand  the  whole  of  it. 

The  examples  are  to  be  performed  in  the  mind,  or  by 
means  of  sensible  objects,  such  as  beans,  nuts,  &c.,  or  by 
means  of  the  plate  at  the  end  of  the  book.  The  pupil  should 
first  perform  the  examples  in  his  own  way,  and  then  be 
made  to  observe  and  tell  how  he  did  them,  and  why  he  did 
them  so.* 

*  It  is  remarkable  that  a  child,  although  he  is  able  to  perform  a  va- 
riety of  examples  which  involve  addition,  subtraction,  multiplication, 
and  division,  recognizes  no  operation  but  addition.  Indeed,  if  we 
analyze  these  operations  when  we  perform  them  in  our  minds,  we 
shall  find  that  they  all  reduce  themselves  to  addition.  They  are  only 
different  ways  of  applying  the  same  principle.  And  it  is  only  when 
we  use  an  artificial  method  of  performing  them,  that  they  take  a 
different  form. 

If  the  following  questions  were  proposed  to  a  child,  his  answers 
would  be,  in  substance,  like  those  annexed  to  the  questions : — How 
much  is  five  less  than  eight  ?  Ans.  Three.  Why  ?  Because  five  and 
three  are  eight.  What  is  the  difference  between  five  and  eight  ?  Ans. 
Three.  Why  1  Because  five  and  three  are  eight.  If  you  divide  eight 
into  two  parts,  such  that  one  of  the  parts  may  be  five,  what  will  the 
other  be  ?  Ans.  Three.  Why  ?  Because  five  and  three  are  eight. 

How  much  must  you  give  for  four  apples,  at  two  cents  apiece  ? 
Ans.  Eight  cents.  Why  ?  Because  two  and  two  are  four,  and  two 
are  six,  and  two  are  eight. 

How  many  apples,  at  two  cents  apiece,  can  you  buy  for  eight 
cents  ?  Ans.  Four.  Why  ?  Because  two  and  two  are  four,  and  two 
are  six,  and  two  are  eight. 

We  shall  be  further  convinced  of  this,  if  we  observe  that  the  same 
table  serves  for  addition  and  subtraction  j  and  another  table,  which  is 


0  PREFACE. 

The  use*  of  the  plates  is  explained  in  the  Key  at  the  end 
of  the  book.  Several  examples  in  each  section  are  performed 
in  the  Key,  to  show  the  method  of  solving  them.  No  answers 
are  given  in  the  book,  except  where  it  is  necessary  to  explain 
something  to  the  pupil.  Most  of  the  explanations  are  given 
in  the  Key  ;  because  pupils  generally  will  not  understand  any 
explanation  given  in  a  book,  especially  at  so  early  an  age. 
The  instructor  must,  therefore,  give  the  explanations  viva  voce. 
These,  however,  will  occupy  the  instructor  but  a  very  short 
time. 

The  first  section  contains  addition  and  subtraction,  the  sec- 
ond multiplication.  The  third  section  contains  division.  In 
tlii.-  section,  the  pupil  learns  the  first  principles  of  fractions,  and 
the  terms  which  are  applied  to  them.  This  is  done  by  making 
him  observe  that  one  is  the  half  of  two,  the  third  of  three,  the 
fourth  of  four,  &c.,  and  that  two  is  two  thirds  of  three,  two 
fourths  of  four,  two  fifths  of  five,  &c. 

The  fourth  section  commences  with  multiplication.  In  this 
the  pupil  is  taught  to  repeat  a  number  a  certain  number  of 
times,  and  a  part  of  another  time.  In  the  second  part  of  this 
section  the  pupil  is  taught  to  change  a  certain  number  of  twos 
into  threes,  threes  into  fours,  &c. 

In  the  fifth  section,  the  pupil  is  taught  to  find  .J,  £,  £,  &c., 
*nd},  1,-f,  &c.,  of  numbers  which  are  exactly  divisible  into 
these  parts.  This  is  only  an  extension  of  the  principle  of 
fractions,  which  is  contained  in  the  third  section. 

In  the  sixth  section,  the  pupil  learns  to  tell  of  what  number 
any  number,  as  2,  3,  4,  &c.,  is  one  half,  one  third,  one  fourth, 
&c. ;  and  also,  knowing  f ,  $,  ^,  &c.,  of  a  number,  to  find  that 
number. 

Tiiese  combinations  contain  all  the  most  common  and  most 
useful  operations  of  vulgar  fractions.  But  being  applied  only 
to  numbers  which  are  exactly  divisible  into  these  fractional 
parts,  the  pupil  will  observe  no  principles  but  multiplication 
and  division,  unless  he  is  told  of  it.  In  fact,  fractions  contain 
no  other  principle.  The  examples  are  so  arranged,  that 
almost  any  child  of  six  or  seven  years  old  will  readily  compre- 
hend them.  And  the  questions  are  asked  in  such  a  manner, 
that,  if  the  instructer  pursues  the  method  explained  in  the. 
Key,  it  will  be  almost  impossible  for  the  pupil  to  perform  any 
example  without  understanding  the  reason  of  it.  Indeed,  in 

formed  by  addition,  serves  both  for  multiplication  and  division.     la 
this  treatise,  the  same  plate  serves  for  the  four  operations. 

This  remark  shows  the  necessity  of  making  the  pupil  attend  to 
his  manner  of  performing  the  examples,  and  of  explaining  to  him  the 
difference  between  the.rn. 


PREFACE.  7 

every  example  which  he  performs,  he  is  obliged  to  go  through 
a  complete  demonstration  of  the  principle  by  which  he 
does  it ;  and  at  the  same  time  he  does  it  in  the  simplest  way 
passible.  These  observations  apply  to  the  remaining  part  of 
the  book. 

These  principles  are  sufficient  to  enable  the  pupil  to  per- 
form almost  all  kinds  of  examples  that  ever  occur.  He  will 
not,  however,  be  able  to  solve  questions  in  which  it  is  neces- 
sary to  take  fractional  parts  of  unity,  though  the  principles 
are  the  same. 

After  section  sixth,  there  is  a  collection  of  miscellaneous 
examples,  in  which  are  contained  almost  all  the  kinds  that 
usually  occur.  There  are  none,  however,  which  the  princi- 
ples explained  are  not  sufficient  to  solve. 

In  section  eighth  and  the  following,  fractions  of  unity  are 
explained,  and,  it  is  believed,  so  simply  as  to  be  intelligible  to 
most  pupils  of  seven  or  eight  years  of  age.  The  operations 
do  not  diiier  materially  from  those  in  the  preceding  sections. 
There  are  some  operations,  however,  peculiar  to  fractions. 
The  two  last  plates  are  used  to  illustrate  fractions. 

When  the  pupil  is  made  familiar  with  all  the  principles  con- 
tained in  this  book,  he  will  be  able  to  perform  all  examples 
in  which  the  numbers  are  so  small,  that  the  operations  may 
be  performed  in  the  mind.  Afterwards,  he  has  only  to  learn 
the  application  of  figures  to  these  operations,  and  his  knowl- 
edge of  arithmetic  will  be  complete. 

The  Rule  of  Three,  and  all  the  other  rules  which  are 
usually  contained  in  our  arithmetics,  will  be  found  useless. 
The  examples  under  these  rules  will  be  performed  upon 
general  principles  with  much  greater  facility,  and  with  a 
greater  degree  of  certainty. 

The  following  are  some  of  the  principal  difficulties  which 
a  child  has  to  encounter  in  learning  arithmetic  in  the  usual 
way,  and  which  are  seldom  overcome  : — First,  the  exam- 
ples are  so  large,  that  the  pupil  can  form  no  conception  of 
the  numbers  themselves ;  therefore  it  is  impossible  for  him 
to  comprehend  the  reasoning  upon  them. — Secondly,  the 
first  examples  are  usually  abstract  numbers.  This  increases 
the  difficulty  very  much ;  for,  even  if  the  numbers  were  so 
small  that  the  pupil  could  comprehend  them,  he  would  dis- 
cover but  very  little  connection  between  them  and  practical 
examples.  Abstract  numbers,  and  the  operations  upon  them, 
must  be  learned  from  practical  examples;  there  is  no  such 
thing  as  deriving  practical  examples  from  those  which  are 
abstract,  unless  the  abstract  have  been  first  derived  from 
those  which  are  practical. — Thirdly,  the  numbers  are  ex- 
pressed by  figures,  which,  if  they  were  used  only  as  a  con- 


8  PREFACE. 

tracted  way  of  writing  numbers,  would  be  much  more  diffi- 
cult to  be  understood  at  first  than  the  numbers  written  at 
length  in  words.  But  they  are  not  used  merely  as  words ; 
they  require  operations  peculiar  to  themselves.  They  are, 
in  fact,  a  new  language,  which  the  pupil  has  to  learn.  The 
pupil,  therefore,  when  he  commences  arithmetic,  is  presented 
with  a  set  of  abstract  numbers,  written  with  figures,  and 
so  large  that  he  has  not  the  least  conception  of  them  even 
when  expressed  in  words.  From  these  he  is  expected  to 
learn  what  the  figures  signify,  and  what  is  meant  by  addition, 
subtraction,  multiplication,  and  division ;  and,  at  the  same 
time,  how  to  perform  these  operations  with  figures.  The 
consequence  is,  that  he  learns  only  one  of  all  these  things,  and 
that  is,  how  to  perform  these  operations  on  figures.  He  can, 
perhaps,  translate  the  figures  into  words ;  but  this  is  useless, 
since  he  does  not  understand  the  words  themselves.  Of  the 
effect  produced  by  the  four  fundamental  operations  he  has 
not  the  least  conception. 

After  the  abstract  examples,  a  few  practical  examples  are 
usually  given;  but  these  again  are  so  large  that  the  pupil 
cannot  reason  upon  them,  and  consequently  he  could  not 
tell  whether  he  must  add,  subtract,  multiply,  or  divide, 
even  if  he  had  an  adequate  idea  of  what  these  operations 
are. 

The  common  method,  therefore,  entirely  reverses  the  natu- 
ral process ;  for  the  pupil  is  expected  to  learn  general  princi- 
ples, before  he  has  obtained  the  particular  ideas  of  which  they 
are  composed. 

The  usual  mode  of  proceeding  is  as  follows  : — The  pupil 
learns  a  rule,  which,  to  the  man  that  made  it,  was  a  general 
principle ;  but  with  respect  to  him,  and  oftentimes  to  the 
instructer  himself,  it  is  so  far  from  it,  that  it  hardly  deserves 
to  be  called  even  a  mechanical  principle.  He  performs  the 
examples,  and  makes  the  answers  agree  with  those  in  the 
book,  and  so  presumes  they  are  right.  He  is  soon  able  to  do 
this  with  considerable  facility,  and  is  then  supposed  to  be 
master  of  the  rule.  He  is  next  to  apply  his  rule  to  practical 
examples ;  but  if  he  did  not  find  the  examples  under  the  rule, 
he  would  never  so  much  as  mistrust  they  belonged  to  it.  But, 
finding  them  there,  he  applies  his  rule  to  them,  and  obtains 
the  answers,  which  are  in  the  book,  and  this  satisfies  him  that 
they  are  right.  In  this  manner  he  proceeds  from  rule  to  rule 
through  the  book. 

When  an  example  is  proposed  to  him,  which  is  not  in  the 
book,  his  sagacity  is  exercised,  not  in  discovering  the  opera- 
tions necessary  to  solve  it,  but  in  comparing  it  with  the  ex- 
amples which  he  has  performed  before,  and  endeavoring  to  dis- 


PREFACE.  9 

cover  some  analogy  between  it  and  them,  either  in  the  sound, 
or  in  something  else.  If  he  is  fortunate  enough  to  discover 
any  such  analogy,  he  finds  what  rule  to  apply,  and  if  he  has 
not  been  deceived  in  tracing  the  analogy,  he  will  probably 
solve  the  question.  His  knowledge  of  the  principles  of  his 
rule  is  so  imperfect,  that  he  would  never  discover  to  which 
of  them  the  example  belongs,  if  he  did  not  trace  it,  by  some 
analogy,  to  the  examples  which  he  had  found  under  it. 

These  observations  do  not  apply  equally  to  all ;  for  some 
will  find  the  right  course  themselves,  whatever  obstacles  be 
thrown  in  their  way.  But  they  apply  to  the  greater  part ; 
and  it  is  probable  that  there  are  very  few  who  have  not  expe- 
rienced more  or  less  inconvenience  from  this  mode  of  proceed- 
ing. Almost  all,  who  have  ever  fully  understood  arithmetic, 
have  been  obliged  to  learn  it  over  again  in  their  own  way. 
And  it  is  not  too  bold  an  assertion  to  say,  that  no  man  ever 
actually  learned  mathematics  in  any  other  method,  than  by 
analytic  induction  ;  that  is,  by  learning  the  principles  by  the 
examples  he  performs ;  and  not  by  learning  principles  first, 
and  then  discovering  by  them  how  the  examples  are  to  be 
performed. 

In  forming  and  arranging  the  several  combinations,  the 
author  has  received  considerable  assistance  from  the  system  of 
Pestalozzi.  He  has  not,  however,  had  an  opportunity  of  seeing 
Pestalozzi's  own  work  on  this  subject,  but  only  a  brief  outline 
of  it  by  another.  The  plates,  also,  are  from  Pestalozzi.  In 
selecting  and  arranging  the  examples  to  illustrate  these  com- 
binations, and  in  the  manner  of  solving  questions  generally, 
he  has  received  no  assistance  from  Pestalozzi. 


THE  BOY  WITHOUT   A   GENIUS. 

Mr.  Wiseman,  the  schoolmaster,  at  the  end  of  his  sum- 
mer vacation,  received  a  new  scholar  with  the  following 
letter  :— 

Sir, — This  will  be  delivered  to  you  by  my  son  Samuel,  whom  I  be? 
leave  to  commit  to  your  care,  hoping  that,  by  your  well-known  skill 
and  attention,  you  will  he  able  to  make  something1  of  him,  which,  I  am 
sorry  to  say.  none  of  his  masters  have  hitherto  done.  He  is  now  elev- 
en, and  yet  can  do  nothing  but  read  his  mother  tongue,  and  that  but 
indifferently.  We  sent  him  at  seven  to  a  grammar  school  in  our 
neighborhood  ;  but  his  master  soon  found  that  his  genius  was  not 
turned  to  learning  languages.  He  was  then  put  to  writing,  but  he 
set  about  it  so  awkwardly  that  he  made  nothing1  of  it.  He  was  tried 
at  accounts,  but  it  appeared  that  he  had  no  genius  for  that  either. 
He  could  clo  nothing  in  geography  for  want  of  memory.  In  short,  if 


10  PREFACE. 

he  has  any  genius  at  all,  it  does  not  yet  show  itself.  But  I  trust  to 
your  experience,  in  cases  of  this  nature,  to  discover  what  he  is  fit  for, 
and  to  instruct  him  accordingly.  I  beg  to  be  favored  shortly  with 
your  opinion  about  him;  and  remain,  sir, 

Your  most  obedient  servant, 

HUMPHREY  ACRES. 

When  Mr.  Wiseman  had  read  this  letter,  he  shook  his  head,  and  said 
to  his  assistant,  A  pretty  subject  they  have  sent  us  here !  a  lad  that  has 
a  great  genius  for  nothing  at  all.  But  perhaps  my  friend  Mr.  Acres 
expects  that  a  boy  should  show  a  genius  for  a  thing  before  he  knows 
any  thing  about  it — no  uncommon  error !  Let  us  see,  however,  what 
the  youth  looks  like.  I  suppose  he  is  a  human  creature  at  least. 

Master  Samuel  Acres  was  now  called  in.  He  came,  hanging  down 
his  head,  and  looking  as  if  he  was  going  to  be  flogged. 

Come  hither,  my  dear  !  said  Mr.  Wiseman.  Stand  by  me,  and  do 
not  be  afraid.  Nobody  will  hurt  you.  How  old  are  you  ? 

Eleven  last  May,  sir. 

A  well-grown  boy  of  your  age,  indeed.  You  love  play,  I  dare  say  7 

Yes,  sir. 

What,  are  you  a  good  hand  at  marbles  ? 

Pretty  good,  sir. 

And  can  spin  a  top  and  drive  a  hoop,  I  suppose  ? 

Yes,  sir. 

Then  you  have  the  full  use  of  your  hands  and  fingers  ? 

Yes,  sir. 

Can  you  write,  Samuel  ? 

I  learned  it  a  little,  sir,  but  I  left  it  off  again. 

And  why  so  ? 

Because  I  could  not  make  the  letters. 

No  !  Why,  how  do  you  think  other  boys  do  ?  Have  they  more 
fingers  than  you  1 

No,  sir. 

Are  you  not  able  to  hold  a  pen  as  well  as  a  marble  ? 

Samuel  was  silent. 

Let  me  look  at  your  hand. 

Samuel  held  out  both  his  paws,  like  a  dancing  bear. 

I  see  nothing  here  to  hinder  you  from  writing  as  well  as  any  boy 
in  the  school.  You  can  read,  I  suppose  ? 

Yes,  sir. 

Tell  me,  then,  what  is  written  over  the  school-room  door. 

Samuel,  with  some  hesitation,  read,  WHATEVER  MAN  HAS 
DONE  MAN  MAY  DO. 

Pray  how  did  you  learn  to  read  ?     Was  it  not  with  taking  pains  ? 

Yes,  sir. 

Well— taking  more  pains  will  enable  you  to  read  better.  Do  you 
know  any  thing  of  the  Latin  Grammar  ? 

No,  sir. 

Have  you  never  learned  it  ? 

I  tried,  sir,  but  I  could  not  get  it  by  heart. 

Why,  you  can  say  some  things  by  heart.  I  dare  say  you  can  tell 
me  the  names  of  the  days  of  the  week  in  their  orrW. 


PREFACE.  11 

Yes.  sir,  I  know  them. 

And  the  months  in  the  year,  perhaps. 

Yes,  sir. 

And  you  could  probably  repeat  the  names  of  your  brothers  and 
sisters,  and  all  your  father's  servants,  and  half  the  people  in  the 
village  besides. 

I  believe  I  could,  sir. 

Well — and  is  hie,  hcec}  hoc,  more  difficult  to  remember  than  these  ? 

Samuel  was  silent. 

Have  you  learned  any  thing  of  accounts  ? 

I  went  into  addition,  sir,  but  I  did  not  go  on  with  it. 

Why  so  ? 

I  could  not  do  it,  sir. 

How  many  marbles  can  you  buy  for  a  penny  1 

Twelve  new  ones,  sir. 

And  how  many  for  a  half-penny  ? 

Six. 

And  how  many  for  two-pence  ? 

Twenty-four. 

If  you  were  to  have  a  penny  a  day,  what  would  that  make  in  a 
week? 

Seven-pence. 

But  if  you  paid  two-pence  out  of  that,  what  would  you  have  left  ? 

Samuel  studied  awhile,  and  then  said,  Five-pence. 

Right.  Why,  here  you  have  been  practising  the  four  great  rules  of 
arithmetic, — addition,  subtraction,  multiplication,  and  division.  Learn- 
ing accounts  is  no  more  than  this.  Well,  Samuel,  I  see  what  you  are 
fit  for.  I  shall  set  you  about  nothing  but  what  you  are  able  to  do  3 
but,  observe,  you  must  do  it.  We  have  no  /  can't  here.  Now  go 
among  your  school-fellows. 

Samuel  went  away,  glad  that  his  examination  was  over,  and  with 
more  confidence  in  his  powers  than  he  had  felt  before. 

The  next  day  he  began  business.  A  boy  less  than  himself  was 
called  out  to  set  him  a  copy  of  letters,  and  another  was  appointed  to 
hear  him  in  grammar.  He  read  a  few  sentences  in  English,  that  he 
could  perfectly  understand,  to  the  master  himself.  'Thus,  by  going 
on  steadily  and  slowly,  he  made  a  sensible  progress.  He  had  already 
joined  his  letters,  got  all  the  declensions  perfectly,  and  half  the  mul- 
tiplication table,  when  Mr.  Wiseman  thought  it  time  to  answer  his 
father's  letter  3  which  he  did  as  follows  :— 

Sir, 

I  now  think  it  right  to  give  you  some  information  concerning 
your  son.  You,  perhaps,  expected  it  sooner ;  but  I  always  wish  to 
avoid  hasty  judgments.  You  mentioned  in  your  letter  that  it  had  not 
yet  been  discovered  which  way  his  genius  pointed.  If  by  genius 
you  meant  such  a  decided  bent  of  mind  to  any  one  pursuit  as  will 
lead  to  excel  with  little  or  no  labor  or  instruction,  I  must  say  that  I 
have  not  met  with  such  a  quality  in  more  than  three  or  four  boys  in 
my  life,  and  your  son  is  certainly  not  among  the  number.  But  if  you 
mean  only  the  ability  to  do  some  of  those  things  which  the  greater  part 


IS)  PREFACE. 

of  mankind  can  do  when  properly  taught,  I  can  affirm  that  I  find  in 
him  no  peculiar  deficiency.  And,  whether  you  choose  to  bring  him 
up  to  trade  or  to  some  practical  profession,  I  see  no  reason  to  doubt 
that  he  may  in  time  become  sufficiently  qualified  for  it.  It  is  my 
favorite  maxim,  sir,  that  every  thing  most  valuable  in  this  life  may 
generally  be  acquired  by  taking  pains  for  it.  Your  son  has  already 
lost  much  time  in  the  fruitless  expectation  of  finding  out  what  he 
would  take  up  of  his  own  accord.  Believe  me,  sir,  few  boys  will 
take  up  any  thing  of  their  own  accord  but  a  top  or  a  marble.  I  will 
take  care,  while  he  is  with  me,  that  he  loses  no  more  time  this  way, 
but  is  employed  about  things  that  are  fit  for  him;  not  doubting  that 
we  shall  find  him  fit  for  them. 

I  am.  sir,  yours,  &c. 

SOLON  WISEMAN. 

Though  the  doctrine  of  this  letter  did  not  perfectly  agree  with  Mr. 
Acres's  notions,  yet,  being  convinced  that  Mr.  Wiseman  was  more 
likely  to  make  something  of  his  son  than  any  of  his  former  preceptors, 
he  continued  him  at  his  school  for  some  years,  and  had  the  satisfaction 
to  find  him  goinff  on  in  a  steady  course  of  gradual  improvement.  In 
due  time,  a  profession  was  chosen  for  him,  which  seemed  to  suit  his 
temper  and  talents,  but  for  which  he  had  no  particular  turn,  having 
never  thought  at  all  about  it.  He  made  a  respectable  figure  in  it,  and 
went  through  the  world  with  credit  and  usefulness,  though  without  a 
genius. 

MRS.  BARBAULD. 


ARITHMETIC. 

PART    I. 

SECTION   I. 

A.*  1.  How  many  thumbs  have  you  on  your  right 
hand  1  How  many  on  your  left  ?  How  many  on 
both  together  1 

2.  How  many  hands  have  you? 

3.  If  you  have  two  nuts  in  one  hand,  and  one  in 
the  other,  how  many  have  you  in  both  ? 

4.  How  many  fingers  have  you  on  one  hand? 

5.  If  you  count  the  thumb  with  the  fingers,  how 
many  will  it  make  ? 

6.  If  you  shut  your  thumb  and  one  finger,  and 
leave  the  rest  open,  how  many  will  be  open  ? 

7.  If  you  have  two  cents  in  one  hand,  and  two  in 
the  other,  how  many  have  you  in  both  ? 

8.  James  has  two  apples,  and  William  has  three  ; 
if  James  gives  his  apples  to  William,  how  many  will 
William  have? 

9.  If  you  count  all  the  fingers  on  one  hand,  and 
two  on  the  other,  how  many  will  there  be? 

10.  George  has  three  cents,  and  Joseph  has  four; 
how  many  have  they  both  together? 

*  For  the  manner  of  solving  questions,  and  the  explanation  of  the 
plates,  see  the  Key  at  the  end  of  the  book.  The  first  questions  in 
this  section  are  intended  for  very  young1  children.  It  will  be  well 
for  the  instmcter  to  give  a  great  many  more  of  this  kind.— -Older 
pupils  may  omit  these. 

2 


14  ARITHMETIC.  [PART  1. 

11.  Robert  gave  five  cents  for  an  orange,  and  two 
for  an  apple;  how  many  did  he  give  for  both? 

12.  If  a  custard  cost  six  cents,  and  an  apple  two 
cents,  how  many  cents  will  it  take  to  buy  an  apple 
and  a  custard  1 

13.  If  you  buy  a  pint  of  nuts  for  five  cents,  and 
an  orange  for  three  cents,  how  many  cents  would 
you  give  for  both?     How  many  more  for  the  nuts 
than  for  the  orange  ? 

14.  If  an  ounce  of  figs  is  worth  six  cents,  and  a 
half  a  pint  of  cherries  is  worth  three  cents,  how 
much  are  they  both  worth  ? 

15.  Dick  had  five  plums,  and  John  gave  him  four 
more ;  how  many  had  he  then  ? 

16.  How  many  fingers  have  you  on  both  hands? 

17.  How  many  fingers  and  thumbs  have  you  on 
both  hands? 

18.  If  you  had  six  marbles  in  one  hand,  and  four 
in  the  other,  how  many  would  you  have  in  the  one 
more  than  in  the  other?     How  many  would  you 
have  in  both  hands? 

19.  David  had  seven  nuts,  and  gave  three  of  them 
to  George ;  how  many  had  he  left  ? 

20.  Two  boys,  James  and  Robert,  played  at  mar- 
bles ;  when  they  began,  they  had  seven  apiece,  and 
when  they  had .  done,  James  had  won  four ;  how 
many  had  each  then  ? 

21.  A  boy,  having  eleven  nuts,  gave  away  three 
of  them ;  how  many  had  he  left  ? 

22.  If  you  had  eight  cents,  and  your  papa  should 
give  you  five  more,  how  many  would  you  have? 

23.  A  man  bought  a  sheep  for  eight  dollars,  and 
a  calf  for    seven    dollars;    what    did    he  give   for 
both? 

24.  A  man  bought  a  barrel  of  flour  for  eight  dol- 
lars, and  sold  it  for  four  dollars  more  than  he  gave 
for  it ;  how  much  did  he  sell  it  for  ? 


SECT.  1.]  ARITHMETIC.  15 

25.  A  man  bought  a  hundred  weight  of  sugar  for 
nine  dollars,  and  a  barrel  of  flour  for  seven  dollars ; 
how  much  did  he  give  for  the  whole  ? 

26.  A  man  bought  three  barrels  of  cider  for  eight 
dollars,  and  ten  bushels  of  apples  for  nine  dollars; 
how  much  did  he  give  for  the  whole  ?      ^  I 

27.  A  man  bought  a  firkin  of  butter  for  twelve 
dollars,  but,  it  being  damaged,  he  sold  it  again  for 
eight  dollars ;  how  much  did  he  lose  ? 

28.  A  man  bought  three  sheep  for  fifteen  dollars, 
but  could  not  sell  them  again  for  so  much  by  eight 
dollars ;  how  much  did  he  sell  them  for  ? 

29.  A  man  bought  sixteen  pounds  of  coffee,  and 
lost  seven  pounds  of  it  as  he  was  carrying  it  home  ; 
how  much  had  he  left  ? 

30.  A  man  bought  nineteen  pounds  of  sugar,  and, 
having  lost  a  part  of  it,  he  found  he  had  nine  pounds 
left ;  how  much  had  he  lost  ? 

31.  A  man,  owing  fifteen  dollars,  paid  nine  dol- 
lars of  it ;  how  much  did  he  then  owe  ? 

32.  A  man,  owing  seventeen  dollars,  paid  all  but 
seven  dollars ;  how  much  did  he  pay  ? 

B.     1.  Two  and  one  are  how  many? 

2.  Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 

4.  Four  and  two  are  how  many  ? 

5.  Five  and  two  are  how  many  ? 

6.  Six  and  two  are  how  many  ? 

7.  Seven  and  two  are  how  many? 

8.  Eight  and  two  are  how  many  ? 

9.  Nine  and  two  are  how  many  ? 

10.  Ten  and  two  are  how  many  ? 

11.  Two  and  three  are  how  many  ? 

12.  Three  and  three  are  how  many? 

13.  Four  and  three  are  how  many  ? 

14.  Five  and  three  are  how  many  ? 


16  ARITHMETIC.  [PART  1. 

15.  Six  and  throe  are  how  many? 

16.  Seven  and  three  are  how  many  1 

17.  Eight  and  three  are  how  many  ? 

18.  Nine  and  three  are  how  many  ? 

19.  Ten  and  three  are  how  many  ] 

20.  Two  and  four  are  how  many? 

21.  Three  and  four  are  how  many  ? 

22.  Four  and  four  are  how  many  ? 

23.  Five  and  four  are  how  many  ? 

24.  Six  and  four  are  how  many? 

25.  Seven  and  four  are  how  many? 

28.  Eight  and  four  are  how  many? 
27.  Nine  and  four  are  how  many  ? 
23.  Ten  and  four  are  how  many  ? 

29.  Two  and  five  are  how  many  ? 

30.  Three  and  five  are  how  many  ? 

31.  Four  and  five  are  how  many  ? 

32.  Five  and  five  are  how  many  ? 

33.  Six  and  five  are  how  many  ? 

34.  Seven  and  five  are  how  many  ? 

35.  Eight  and  five  are  how  many  ? 

36.  Nine  and  five  are  how  many  ? 

37.  Ten  and  five  are  how  many  ? 
33.  Two  and  six  are  how  many? 

39.  Three  and  six  are  how  many  ? 

40.  Four  and  six  are  how  many  ? 

41.  Five  and  six  are  how  many  ? 

42.  Six  and  six  are  how  many  ? 

43.  Seven  and  six  are  how  many  ? 

44.  Eight  and  six  are  how  many  ? 

45.  Nine  and  six  are  how  many  ? 

46.  Ten  and  six  are  how  many  ? 

47.  Two  and  seven  are  how  many  ? 

48.  Three  and  seven  are  how  many  ? 

49.  Four  and  seven  are  how  many  ? 

50.  Five  and  seven  are  how  many? 

51.  Six  and  seven  are  how  many? 


SECT.  1.]  ARITHMETIC.  17 

52.  Seven  and  seven  are  how  many  ? 

53.  Eight  and  seven  are  how  many  1 

54.  Nine  and  seven  are  how  many  ? 

55.  Ten  and  seven  are  how  many  1 

56.  Two  and  eight  are  how  many  ? 

57.  Three  and  eight  are  how  many  ? 

58.  Four  and  eight  are  how  many  1 

59.  Five  and  eight  are  how  many  ? 

60.  Six  and  eight  are  how  many  1 

61.  Seven  and  eight  are  how  many? 

62.  Eight  and  eight  are  how  many  1 

63.  Nine  and  eight  are  how  many  ? 

64.  Ten  and  eight  are  how  many  ? 

65.  Two  and  nine  are  how  many? 

66.  Three  and  nine  are  how  many  1 

67.  Four  and  nine  are  how  many  ? 

68.  Five  and  nine  are  how  many  ? 

69.  Six  and  nine  are  how  many  ? 

70.  Seven  and  nine  are  how  many  ? 

71.  Eight  and  nine  are  how  many? 

72.  Nine  and  nine  are  how  many  ? 

73.  Ten  and  nine  are  how  many  ? 

74.  Two  and  ten  are  how  many  ? 

75.  Three  and  ten  are  how  many  ? 

76.  Four  and  ten  are  how  many  1 

77.  Five  and  ten  are  how  many  ? 

78.  Six  and  ten  are  how  many  ? 

79.  Seven  and  ten  are  how  many  ? 

80.  Eight  and  ten  are  how  many  ? 

81.  Nine  and  ten  are  how  many? 

82.  Ten  and  ten  are  how  many  ? 

C.  1.  Two  and  one  are  how  many  ? 

2.  Two  and  two  are  how  many  ? 

3.  Three  and  two  are  how  many  ? 

4.  Five  und  two  are  how  many  1 

5.  Four  and  two  are  how  many  ? 

2* 


18  ARITHMETIC.  [PART  1. 

6.  Six  and  two  are  how  many  ? 

7.  Eight  and  two  are  how  many  ? 

8.  Five  and  three  are  how  many  ? 

9.  Seven  and  three  are  how  many  ? 

10.  Four  and  three  are  how  many  ? 

11.  Two  and  three  are  how  many  ? 

12.  Two  and  six  are  how  many  ? 

13.  Two  and  eight  are  how  many  ? 

14.  Six  and  three  are  how  many  ? 

15.  Three  and  four  are  how  many  ? 

16.  Three  and  six  are  how  many  ? 

17.  Two  and  seven  are  how  many  ? 

18.  Ten  and  two  are  how  many  ? 

19.  Two  and  four  are  how  many? 

20.  Three  and  seven  are  how  many? 

21.  Four  and  four  are  how  many  ? 

22.  Five  and  four  are  how  many  ? 

23.  Seven  and  two  are  how  many  ? 

24.  Two  and  five  are  how  many  ? 

25.  Three  and  three  are  how  many  ? 

26.  Four  and  five  are  how  many  ? 

27.  Nine  and  two  are  how  many  ? 

28.  Three  and  five  are  how  many  ? 

29.  Two  and  ten  are  how  many  7 

30.  Three  and  eight  are  how  many  ? 

31.  Ten  and  three  are  how  many  ? 

32.  Two  and  nine  are  how  many  ? 

33.  Four  and  six  are  how  many  ? 

34.  Eight  and  three  are  how  many  ? 

35.  Seven  and  four  are  how  many  ? 

36.  Nine  and  three  are  how  many  ? 

37.  Six  and  four  are  how  many  ? 

38.  Five  and  five  are  how  many  ? 

39.  Three  and  nine  are  how  many  ? 

40.  Four  and  seven  are  how  many  ? 

41.  Six  and  five  are  how  many? 

42.  Three  and  ten  are  how  many  ? 


SECT.  1.]  ARITHMETIC.  19 

43.  Eight  and  four  are  how  many  1 

44.  Five  and  eight  are  how  many  ? 

45.  Four  and  nine  are  how  many  ? 

46.  Five  and  six  are  how  many  ? 

47.  Ten  and  four  are  how  many  ? 

48.  Seven  and  five  are  how  many  1 

49.  Six  and  six  are  how  many  1 

50.  Nine  and  four  are  how  many? 

51.  Eight  and  five  are  how  many  ? 

52.  Five  and  nine  are  how  many  ? 

53.  Four  and  ten  are  how  many  ? 

54.  Six  and  seven  are  how  many  ? 

55.  Four  and  eight  are  how  many  ? 

56.  Nine  and  five  are  how  many  ? 

57.  Six  and  eight  are  how  many  ? 

58.  Ten  and  five  are  how  many  ? 

59.  Seven  and  six  are  how  many  1 

60.  Eight  and  seven  are  how  many  ? 

61.  Six  and  nine  are  how  many  ? 

62.  Seven  and  seven  are  how  many  ? 

63.  Eight  and  six  are  how  many  1 

64.  Ten  and  six  are  how  many  ? 

65.  Eight  and  eight  are  how  many  ? 

66.  Nine  and  seven  are  how  many  ? 

67.  Ten  and  eight  are  how  many  ? 

68.  Six  and  ten  are  how  many  ? 

69.  Five  and  seven  are  how  many  ? 

70.  Nine  and  six  are  how  many  ? 

71.  Seven  and  eight  are  how  many  ? 

72.  Eight  and  nine  are  how  many  ? 

73.  Nine  and  nine  are  how  many  ? 

74.  Five  and  ten  are  how  many  ? 

75.  Seven  and  nine  are  how  many  ? 

76.  Nine  and  eight  are  how  many  T 

77.  Eight  and  ten  are  how  many  ? 

78.  Ten  and  nine  are  how  many  ? 

79.  Seven  and  ten  are  how  many  ? 


20  ARITHMETIC.  [PART  1. 

80.  Nine  and  ten  are  how  many  ? 

81,  Ten  and  ten  are  how  many  ? 

D.  1.  Three  boys,  Peter,  John,  and  Oliver,  gave 
some  money  to  a  beggar ;  Peter  gave  seven  cents, 
John  four  cents,  and  Oliver  three  cents ;  how  many 
did  they  all  give  him  ? 

2.  How  many  did  Peter  give  more  than  Oliver  ? 

3.  Frank  had  nine  pears,  and  gave  three  of  them 
to  Harry ;  how  many  had  he  left  ?  and  how  many 
more  than  Harry  had  he  then  ? 

4.  Dick    had    ten    peaches,  Harry    twelve,   and 
Charles  thirteen ;    Dick    gave    three    to    Stephen, 
Harry  gave  him  six,  and  Charles   gave  him  five  ; 
how  many  had  Stephen  ?  and  how  many  had  each 
left? 

5.  A  boy  had  twenty  apples,  and  gave  them  to 
his  companions  as  follows:  to  one  he  gave  three; 
to  another,  two ;  to  another,  four ;  and  to  another, 
five  :  how  many  did  he  give  away  ?  and  how  many 
had  he  left  ? 

6.  A  boy  gave  to  one  of  his  companions  eight 
peaches ;    to  another,  six ;    to   another,  four ;    and 
kept  two  himself;  how  many  had  he  at  first  ? 

7.  A  boy  went  to  the  confectioner's  and  bought 
three  cakes  of  gingerbread,  for  which  he  gave  a 
cent  apiece  ;  two  bunns,  for  which  he  gave  three 
cents  apiece ;  one  custard  for  four  cents,  and  one 
orange  for  six  cents ;  how  many  cents  did  he  spend 
for  the  whole  ? 

8.  A  boy,  having  twenty-five  cents,  bought  one 
quart  of  cherries  for  eight  cents,  one   orange    for 
six  cents,  and  gave  away  three  cents ;  how  many 
cents  had  he  left  ? 

9.  A  boy  bought  a  box  for  eighteen  cents,  and  gave 
eight  cents  to  have  it  painted,  and  then  sold  it  for  thir- 
ty-two cents ;  how  much  did  he  gain  by  the  bargain  ? 


SECT.  1.]  ARITHMETIC.  21 

10.  A  man  bought  a  sleigh  for  seventeen  dollars, 
and  gave  nine  dollars  to  have  it  repaired  and  paint- 
ed, and  then  sold  it  for  twenty-three  dollars ;  how 
much  did  he  lose  by  the  bargain? 

11.  Eleven  and  two  are  how  many? 
1*2.  Eleven  and  three  are  how  many  ? 

13.  Eleven  and  four  are  how  many  ? 

14.  Eleven  and  five  are  how  many  ? 

15.  Eleven  and  six  are  how  many? 
1C.  Eleven  and  seven  are  how  many  ? 

17.  Eleven  and  eight  are  how  many  ? 

18.  Eleven  and  nine  are  how  many  ? 

19.  Eleven  and  ten  are  how  many  ? 

20.  Twelve  and  two  are  how  many? 

21.  Twelve  and  three  are  how  many? 

22.  Twelve  and  four  are  how  many  ? 

23.  Twelve  and  five  are  how  many  ? 

24.  Twelve  and  six  are  how  many  ? 

25.  Twelve  and  seven  are  how  many  ? 

26.  Twelve  and  eight  are  how  many  ? 

27.  Twelve  and  nine  are  how  many  ? 

28.  Twelve  and  ten  are  how  many? 

29.  Thirteen  and  two  are  how  many  ? 

30.  Thirteen  and  three  are  how  many  ? 

31.  Thirteen  and  four  are  how  many  ? 

32.  Thirteen  and  five  are  how  many? 

33.  Thirteen  and  six  are  how  many  ? 

34.  Thirteen  and  seven  are  how  many  ? 

35.  Fourteen  and  two  are  how  many  ? 

36.  Fourteen  and  three  are  how  many  ? 

37.  Fourteen  and  four  are  how  many? 

38.  Fourteen  and  five  are  how  many  ? 

39.  Fourteen  and  six  are  how  many? 

40.  Fifteen  and  two  are  how  many  ? 

41.  Fifteen  and  three  are  how  many? 

42.  Fifteen  and  four  are  how  many? 


22  ARITHMETIC.  [PART  1. 

43.  Fifteen  and  five  are  how  many  ? 

44.  Sixteen  and  two  are  how  many  ? 

45.  Sixteen  and  three  are  how  many? 

46.  Sixteen  and  four  are  how  many? 

47.  Seventeen  and  two  are  how  many  ? 

48.  Seventeen  and  three  are  how  many  ? 

49.  Eighteen  and  two  are  how  many  ? 

E.  1.  A  man  bought  a  sheep  for  nine  dollars,  and 
to  pay  for  it  he  gave  five  bushels  of  corn  worth  four 
dollars,  and  the  rest  in  money ;  how  much  money 
did  he  pay  ? 

2.  If  a  barrel  of  flour  is  worth  eight  dollars,  and 
a  hundred  weight  of  sugar  is  worth  twelve  dollars, 
how  much  more  is  the  sugar  worth  than  the  flour  ? 

3.  If  a  man  had  eleven  dollars,  and  should  buy 
three  bushels  of  corn  for  five  dollars,  how  much 
money  would  he  have  left  ? 

4.  A  man  bought  a  firkin  of  butter  for  fifteen  dol- 
ars,  but,  it  being  damaged,  he  was  willing  to  sell  it 

again  for  eight  dollars  less  than  he  gave  for  it ;  what 
did  he  sell  it  for  ? 

5.  A  man  bought  three  barrels  of  flour  for  eighteen 
dollars,  and  sold  it  again  for  eleven  dollars ;  what 
did  he  lose  by  the  bargain  ? 

6.  A  man  bought  a  keg  of  tobacco  for  thirteen 
dollars,  and  sold  it  again  for  eighteen ;  what  did  he 
gain  by  the  bargain  ? 

7.  Five  less  two  are  how  many  ? 

8.  Seven  less  three  are  how  many  ? 

9.  Three  less  three  are  how  many  ? 

10.  Nine  less  three  are  how  many? 

11.  Six  less  two  are  how  many? 

12.  Seven  less  four  are  how  many  ? 

13.  Eight  less  three  are  how  many  ? 

14.  Five  less  four  are  how  many  ? 


SECT.  1.]  ARITHMETIC.  23 

15.  Seven  less  five  are  how  many  ? 

16.  Nine  less  five  are  how  many  ? 

17.  Eight  less  six  are  how  many? 

18.  Eleven  less  two  are  how  many  ? 

19.  Twelve  less  four  are  how  many  1 

20.  Ten  less  seven  are  how  many  ? 

21.  Thirteen  less  five  are  how  many  1 

22.  Fourteen  less  eight  are  how  many  ? 

23.  Twelve  less  seven  are  how  many  ? 

24.  Seventeen  less  five  are  how  many  ? 

25.  Eighteen  less  ten  are  how  many  ? 

26.  Thirteen  less  seven  are  how  many  ? 

27.  Sixteen  less  seven  are  how  many  ? 

28.  Fifteen  less  seven  are  how  many  ? 

29.  Nineteen  less  six  are  how  many  ? 

30.  Eighteen  less  five  are  how  many  ? 

31.  Seventeen  less  eight  are  how  many? 

32.  Fourteen  less  nine  are  how  many  ? 

33.  Sixteen  less  five  are  how  many  ? 

34.  Fifteen  Jess  eight  are  how  many? 

35.  Fourteen  less  nine  are  how  many  ? 

36.  Sixteen  less  ten  are  howT  many  ? 

37.  Seventeen  less  nine  are  how  many  ? 

38.  Eighteen  less  seven  are  how  many  ? 

F.  1.  How  many  are  nine  and  two?  Nineteen 
and  two  1  Twenty-nine  and  two  ?  Thirty-nine  and 
two  ?  Forty-nine  and  two  ?  Fifty-nine  and  two  ?  Six- 
ty-nine and  two  ?  Seventy-nine  and  two  ?  Eighty- 
nine  and  two  ?  Ninety-nine  and  two  ? 

2.  How  many    are   nine   and   three  ?     Nineteen 
and  three  ?    Twenty-nine  and   three  ?    Thirty-nine 
and  three  ?  Forty-nine  and  three  1    Fifty-nine  and 
three  ?     Sixty-nine  and   three  ?     Seventy-nine  and 
three  ?     Eighty-nine  and  three  ?     Ninety-nine  and 
three  ? 

3.  How  many  are  nine  and  four  ?    Nineteen  and 


24  ARITHMETIC.  [PART  1. 

four  ?  Twenty-nine  and  four  ?  Thirty-nine  and  four  ? 
Forty-nine  and  four?  Fifty-nine  and  four?  Sixty- 
nine  and  four  ?  Seventy-nine  and  four  ?  Eighty-nine 
and  four  ?  Ninety-nine  and  four  ? 

4.  How  many  are  nine  and  five  ?     Nineteen  and 
five  ?  Twenty-nine  and  five  ?  Thirty-nine  and  five  ? 
Forty-nine  and  five  ?     Fifty-nine  and  five  ?     Sixty- 
nine  and  five  ?  Seventy-nine  and  five  ?    Eighty-nine 
and  five  ?  Ninety-nine  and  five  ? 

5.  How  many  are  nine  and  six  ?     Nineteen  and 
six?    Twenty-nine  and  six?    Thirty-nine  and  six? 
Forty-nine  and  six?    Fifty-nine  and  six?  Sixty-nine 
and  six  ?     Seventy-nine  and  six  ?     Eighty-nine  and 
six  ?  Ninety-nine  and  six  ? 

6.  How  many  are  nine  and  seven  ?  Nineteen  and 
seven  ?    Twenty-nine  and  seven  ?    Thirty-nine  and 
seven  ?  Forty-nine  and  seven  ?  Fifty-nine  and  seven  ? 
Sixty-nine  and  seven  ?     Seventy-nine    and   seven  ? 
Eighty-nine  and  seven  ?  Ninety-nine  and  seven  ? 

7.  How  many  are  nine  and  eight  ?  Nineteen  and 
eight  ?    Twenty-nine  and  eight  ?    Thirty-nine  and 
eight  ?  Forty-nine  and  eight  ?  Fifty-nine  and  eight  ? 
Sixty-nine    and   eight  ?     Seventy-nine    and   eight  ? 
Eighty-nine  and  eight  ?  Ninety-nine  and  eight  ? 

8.  How  many  are  nine  and  nine  ?  Nineteen  and 
nine  ?     Twenty-nine  and  nine  ?     Thirty-nine  and 
nine?     Forty-nine  and  nine ?  Fifty-nine  and  nine  ?' 
Sixty-nine   and   nine  ?      Seventy-nine    and   nine  ? 
Eighty-nine  and  nine  ?  Ninety-nine  and  nine  ? 

9.  How  many  are  nine  and  ten  ?    Nineteen  and 
ten  ?  Twenty-nine  and  ten  ?    Thirty-nine  and  ten  ? 
Forty-nine  and  ten  ?  Fifty-nine  and  ten  ?  Sixty-nine 
and  ten  ?     Seventy-nine  and  ten  ?    Eighty-nine  and 
ten  ?  Ninety-nine  and  ten  ? 

10.  How  many  are  eight  and  three  ?  Eighteen  and 
three  ?  Twenty-eight  and  three  ?    Thirty-eight  and 
three  ?     Forty-eight   and   three  ?     Fifty-eight  and 


SECT.  1.]  ARITHMETIC.  25 

three  1  Sixty-eight  and  three  ?  Seventy-eight  and 
three  ?  Eighty-eight  and  three  ?  Ninety-eight  and 
three  ? 

11.  How  many  are  eight   and   four?     Eighteen 
and  four  ?     Twenty-eight  and  four  ?     Thirty-eight 
and  four  ?     Forty-eight  and  four  1     Fifty-eight  and 
four  ?    .Sixty-eight   and  four  ?     Seventy-eight    and 
four?     Eighty-eight   and   four?     Ninety-eight  and 
four? 

12.  How  many  are  eight  and  five  ?  Eighteen  and 
five  ?     Twenty-eight   and   five  ?     Thirty-eight  and 
five  ?     Forty-eight  and  five  ?     Fifty-eight  and  five  ? 
Sixty-eight    and    five  ?-   Seventy-eight    and    five  ? 
Eighty-eight  and  five  ?    Ninety-eight  and  five  ? 

13.  How  many  are  eight  and  six?  Eighteen  and 
six?  Twenty-eight  and  six?  Thirty-eight  and  six? 
Forty-eight  and  six  ?     Fifty-eight  and  six  ?     Sixty- 
eight  and  six  ?  Seventy-eight  and  six  ?  Eighty-eight 
and  six  ?  Ninety-eight  and  six  ?. 

14.  How  many  are  eight  and  seven  ?     Eighteen 
and  seven  ?  Twenty-eight  and  seven  ?    Thirty-eight 
and  seven  ?  Forty-eight  and  seven  ?  Fifty-eight  and 
seven  ?     Sixty-eight  and  seven  ?    Seventy-eight  and 
seven  ?    Eighty-eight  and  seven  ?    Ninety-eight  and 
seven  ? 

15.  How  many  are  eight  and  eight?     Eighteen 
anci  eight  ?    Twenty-eight  and  eight  ?    Thirty-eight 
and  eight?       Forty-eight  and  eight?      Fifty-eight 
and  eight  ?     Sixty-eight  and  eight  ?     Seventy-eight 
and  eight  ?  Ninety-eight  and  eight  ? 

16.  How  many  are  eight  and  nine  ?  Eighteen  and 
nine  ?     Twenty-eight  and  nine  ?     Thirty-eight  and 
nine  ?  Forty-eight  and  nine  ?  Fifty-eight  and  nine  ? 
Sixty-eight    and    nine  ?     Seventy-eight   and  nine  ? 
Eighty-eight  and  nine?  Ninety-eight  and  nine? 

17.  How  many  are  seven  and  four  ?    Seventeen 
and  four  ?     Twenty-seven  and  four  ?     Thirty-seven 

3 


36  ARITHMETIC.  [PART  1. 

and  four  ?  Forty-seven  and  four  ?  Fifty-seven  and 
four  ?  Sixty-seven  and  four  ?  Seventy-seven  and 
four  ?  Eighty-seven  and  four  1  Ninety-seven  and 
four  ? 

18.  How  many  are  seven  and  five  ?    Seventeen 
and  five  ?     Twenty-seven  and  five  ?     Thirty-seven 
and  five  ?     Forty-seven  and  five  1     Fifty-seven  and 
five  ?  Sixty-seven  and  five  ?  Seventy-seven  and  five  ? 
Eighty-seven  and  five  ?  Ninety-seven  and  five  ? 

19.  How  many  are  seven  and  six?     Seventeen 
and  six  ?  Twenty-seven  and  six  ?  Thirty-seven  and 
six?  Forty-seven  and  six  ?  Fifty-seven  and  six?  Six- 
ty-seven and  six  ?    Seventy-seven  and  six  ?'  Eighty- 
seven  and  six  ?  Ninety-seven  and  six  ? 

20.  How  many  are  seven  and  seven  ?    Seventeen 
and  seven  ?  Twenty-seven  and  seven  ?  Thirty-seven 
and*  seven  ?  Forty-seven  and  seven  ?  Fifty-seven  and 

% seven  ?  Sixty-seven  and  seven  ?  Seventy-seven  and" 
seven  ?  Eighty-seven  and  seven  ?  Ninety-seven  and 
seven  ? 

21.  How  many  are  seven  and  eight?     Seventeen 
and  eight  ?  Twenty-seven  and  eight  ?  Thirty-seven 
and  eight  ?  Forty-seven  and  eight  \  Fifty-seven  and 
eight  ?    Sixty-seven  and  eight  ?    Seventy-seven  and 
eight  ?  Eighty-seven  and  eight  ?    Ninety-seven  and 
eight? 

22.  How  many  are  seven  and  nine  ?    Seventeen 
and  nine  ?    Twenty-seven  and  nine  ?    Thirty-seven 
and  nine  ?    Forty-seven  and  nine  ?    Fifty-seven  and 
nine  ?     Sixty-seven  and  nine  ?     Seventy-seven  and 
nine  ?     Eighty-seven  and  nine  ?     Ninety-seven  and 
nine? 

23.  How  many  are  six  and  five  ?  Sixteen  and  five  ? 
Twenty-six  and  five ?    Thirty-six  and  five  1    Forty-, 
six  and  five?  Fifty-six  arid  five?  Sixty-six  and  five? 
Seventy-six  and  five  ?  Eighty-six  and  five  ?  Ninety- 
six  and  five  7 


SECT.  1.]  ARITHMETIC.  27 

24.  How  many  are  six  and  six  ?  Sixteen  and  six  ? 
Twenty-six  and  six  ?  Thirty-six  and  six  ?  Forty-six 
and  six  ?  Fifty-six  and  six  ?  Sixty-six  and  six  ?  Sev- 
enty-six and  six?    Eighty-six  and  six?    Ninety-six 
and  six  ? 

25.  How  many  are  six  and  seven  ?     Sixteen  and 
seven  ?  Twenty-six  and  seven  ?  Thirty-six  and  sev- 
en ?  Forty-six  and  seven  ?  Fifty^six  and  seven  1  Six- 
ty-six and  seven  ?  Seventy-six  and  seven  ?  Eighty- 
six  and  seven  1  Ninety-six  and  seven  1 

26.  How  many  are  six  and  eight  ?     Sixteen  and 
eight  ?  Twenty-six  and  eight  ?  Thirty-six  and  eight  1 
Forty-six  and  eight  1  Fifty-six  and  eight  ?    Sixty-six 
and  eight  1  Seventy-six  and  eight  ?    Eighty-six  and 
eight  1  Ninety-six  and  eight  ? 

27.  How  many  are  six  and  nine?     Sixteen  and 
nine  ?  Twenty-six  and  nine  ?  Thirty-six  and  nine  ? 
Forty-six  and  nine?  Fifty-six  and  nine?  Sixty-six 
and  nine  ?    Seventy-six  and  nine  ?    Eighty-six  and 
nine  ?  Ninety-six  and  nine  ? 

28.  How  many  are  five  and  six  ?  Fifteen  and  six  ? 
Twenty-five  and  six  ?    Thirty-five  and  six  ?    Forty- 
five  and  six?  Fifty-five  and  six  ?  Sixty-five  and  six? 
Seventy-five  and  six  ?    Eighty-five  and  six  ?  -Ninety- 
five  and  six  ? 

29.  How  many  are  five  and  seven  ?  Fifteen  and 
seven  ?     Twenty-five  and  seven  ?     Thirty-five  and 
seven  ?  Forty-five  and  seven  ?  Fifty-five  and  seven  ? 
Sixty-five  and    seven  ?      Seventy-five    and    seven  ? 
Eighty-five  and  seven  ?  Ninety-five  and  seven  ? 

30.  How  many  are  five  and  eight  ?    Fifteen  and 
eight  ?     Twenty-five  and    eight  ?     Thirty-five    and 
eight?  Forty-five  and  eight?    Fifty-five  and  eight? 
Sixty-five  and  eight  ?  Seventy-five  and  eight  ?  Eighty- 
five  and  eight  ?  Ninety-five  and  eight  ? 

31.  How  many  are  five  and  nine  ?     Fifteen  and 
nine  ?  Twenty-five  and  nine  ?  Thirty-five  and  nine  ? 


28  ARITHMETIC.  [PART  1 

Forty-five  and  nine  ?  Fifty-five  and  nine  ?  Sixty-five 
and  nine  ?  Seventy-five  and  nine  ?  Eighty -five  and 
nine  ?  Ninety-five  and  nine  ? 

32.  How  many  are  four  and  seven  ?  Fourteen  and 
seven  ?     Twenty-four  and  seven  ?     Thirty-four  and 
seven  ?  Forty-four  and  seven  ?  Fifty-four  and  seven  ? 
Sixty-four   arid   seven  ?      Seventy-four    and  seven  ? 
Eighty-four  and  seven  ?  Ninety-four  and  seven  '? 

33.  How  many  are  four  and  eight  ?  Fourteen  and 
eight  1     Twenty-four  and  eight  ?     Thirty-four  and 
eight  ?  Forty-four  and  eight  ?  Fifty-four  and  eight  ? 
Sixty-four  and  eight  1      Seventy-four    and    eight  ? 
Eighty-four  and  eight?  Ninety-four  and  eight  I 

34.  How  many  are  four  and  nine  1  Fourteen  and 
nine  ?  Twenty-four  and  nine  ?  Thirty-four  and  nine  1 
Forty-four  and  nine?    Fifty-four  and  nine?    Sixty- 
four  and  nine  ?  Seventy-four  and  nine  ?  Eighty-four 
and  nine  ?  Ninety-four  and  nine  ? 

35.  How  many  are  three  and  eight  ?     Thirteen 
and  eight  ?    Twenty-three  and  eight  ?    Thirty-three 
and  eight  ?  Forty-three  and  eight  ?    Fifty-three  and 
eight?    Sixty-three  and  eight?    Seventy-three  and 
eight  ?     Eighty-three  and  eight  ?    Ninety-three  'and 
eight  ? 

36.  How  many  are  three  and  nine  ?  Thirteen  and 
nine  ?     Twenty-three  and  nine  ?     Thirty-three  and 
nine  ?  Forty-three  and  nine  ?  Fifty-three  and  nine  ? 
Sixty-three   and   nine  ?      Seventy-three  and  nine  ? 
Eighty-three  and  nine  ?  Ninety-three  and  nine  ? 

37.  How  many  are  two  and  nine  ?     Twelve  and 
nine  ?  Twenty-two  and  nine  ?  Thirty-two  and  nine  ? 
Forty-two  and  nine  ?    Fifty-two  and  nine  ?      Sixty- 
two  and  nine  ?  Seventy-two  and  nine  ?  Eighty-two 
and  nine  ?  Ninety-two  and  nine  ? 

Cr.   1.  A  man  bought  a  firkin  of  butter  for  nine 
dollars,  a  keg  of  molasses  for  six  dollars,  and  five 


SECT.  1.]  ARITHMETIC.  29 

bushels  of  wheat  for  seven  dollars ;  how  much  did 
he  give  for  the  whole  1  v 

2.  .A  boy  gave  some  apples  to  his  companions ;  to 
one  he  gave  seven,  to  another  six,  and  to  another 
eight ;  how  many  did  he  give  to  the  whole  ? 

3.  A  man  bought  a  cow  for  seventeen  dollars,  a 
sheep  for  nine,  and  a  calf  for  seven  ;  how  much  did 

•  he  give  for  the  whole  ? 

4.  A  drover  bought  sheep  as  follows ;  of  one  man 
he  bought  twenty-seven,  of  another  eight,  of  anoth- 
er  ten,  and   of  another   five ;    afterwards   he   sold 
nine  of  them  ;  how  many  had  he  then  ? 

5.  A  lady  bought  a  comb  for  thirty-seven  cents, 
some  tape  "for  eight  cents,  some  pins  for   ten  cents, 
some  needles    for  six  cents,   and   some  thread  for 
six  cents  ;  she  gave  seventy-five  cents ;  how  much 
change  ought  she  to  receive  back  ? 

6.  Eight,  and  nine,  and  six,  are  how  many  ? 

7.  Five,  and  seven,  and  three,  are  how  many  1 

8.  Four,  and  three,  less  two,  are  how  many  ? 

9.  Seven,  and  five,  less  three,  are  how  many  ? 

10.  Sixteen,  and  nine,  and  three,  are  how  many  ? 

11.  Twenty-three  and  eight  are  how  many? 

12.  Twenty-seven  and  five  are  how  many  ? 

13.  Twenty-five,  less  eight,  are  how  many? 

'  14.  Thirty-two    and    seven,  less   nine,  are    how 
many? 

15.  Thirty-eight,   and  six,  and  four,  less   seven, 
are  how  many  ? 

16.  Forty-four,  and  eight,  and  'three,  and  seven, 
are  how  many  1 

17.  Fifty-two,  and  six,  and  four,  and  five,    and 
three,  are  how  many  ? 

18.  Fifty-seven,  and  six,  and  three,  and  five,  and 
two,  less  eight,  are  how  many  ? 

19.  Sixty-three,  and  five,  and  four,  and  six,  and 
two,  less  seven,  are  how  many  ? 

3* 


30  ARITHMETIC.  [PART  1. 

20.  Seventy-five,  and  six,  and  eight,  and  three, 
and  seven,  and  four,  less  nine*,  are  how  many  ? 

21.  Eighty-three,  and  six,  and  five,  and  two,  and 
seven,  and  nine,  less  four,  are  how  many  ? 

22.  Fifty-eight,  rand  ten,  and  five,  and  seven,  and 
three,  and  six,  and  four,  less  nine,  are  how  many  ? 

23.  Sixty-seven,  and  five,  and  e'ight,  and  nine, 
and  seven,  less  six,  are  how  many  ? 

24.  Seventy-four,  and  nine,  and  seven,  and  five, 
and  two,  less  six,  are  how  many  ? 

25.  Seventy-eight,  and  seven,  and  six,  and  two, 
and  five,  and  eight,  less  nine,  are  how  many  ? 

26.  Eighty-four,  and  seven,  and  six,  and    eight, 
and  five,  less  ten,  are  how  many  ? 

*27.  Forty-seven,  and  eight,  and  six,  and  two,  and 
four,  and  eight,  and  three,  and  seven,  and  ten,  and 
nine,  less  five,  are  how  many  ? 

28.  Thirty-five,  and  eight,  and  four,  and  six,  and 
three,  and  four,  less  eleven,  are  how  many  ? 

29.  Seventy,    and   ten,  and   six,  and   nine,  ^.nd 
seven,  and  two,  and  five,  and  eight,  and  nine,  less 
three,  are  how  many  ?  ^* 

H.  1.  A  man  bought  a  cow  for  twenty-eight 
dollars',  and  a  sheep  for  four  dollars,  and  a  pig  for 
seven  dollars ;  how  much  did  he  give  for  the  whole  ? 

2.  James    had    twenty-seven  cents;   John    gave 
him  four  more,  David  seven,  and  George  eleven, 
and   he  bought  nine   cents'    worth   of  cake;    how 
many  cents  had  lie  left  ? 

3.  A  man  paid  sixteen  dollars  to  A,  nine  dollars 
to  B,  seven  dollars  to  C,  ten  dollars  to  D,  six  dol- 
lars to  E,  four  dollars  to  F,  and  had  eight  dollars 
left ;  how  many  had  he  at  first  ? 

4.  From  Boston  to  Roxbury  it  is  three  miles ;  from 
Roxbury  to  Dedham,  six  miles ;  from  Dedham  to  Wai- 
pole,  eleven  miles ;  from  Walpole  to  Wrentham.,  four 


SECT.  2.]  ARITHMETIC.  31 

miles ;  from  Wrentham  to  Attleborough,  four  miles ; 
from  Attleborough  to  Pawtucket,  nine  miles;  from 
Pawtucket  to  Providence,  four  miles :  how  many 
miles  is  it  from  Boston  to  Providence  1 

5.  One  boy  had  fifteen  nuts ;   another  boy  gave 
him  seven ;  another  nine ;   and  another    gave  him 
enough  to  make  his  number  forty :  how  many  did 
the  last  boy  give  him  ? 

6.  A  l)oy  had  thirty-seven  apples  :  he  gave  five  to 
one  companion,  and  eight  to  another  ;  and  when  he 
had  given  some  to    another,  he  had  six  left :  how 
many  did  he  give  to  the  last  ? 

7.  A  man  owed  fifty-six  dollars  :  at  one  time  he 
paid  sevenfeen  dollars  ;  at  another,  eight ;  at  another, 
five  ;  at  another,  seven  ;  at  last  he  paid  the  rest  of 
the  debt,  wanting  four  dollars:  how  much  .was  the 
last  payment  ? 

8.  Six  men  bought  a  horse  for  seventy  dollars :  the 
first  gave  twenty-three  dollars  ;    the  second,  fifteen ; 
the  third,  twelve ;  the  fourth,  nine ;  the  fifth,  seven : 
how  much  did  the  sixth  give  ? 

9.  A  man   bought  a  horse  for  forty-five  dollars, 
and  paid  fifteen  dollars  for  keeping  him ;    he  let  him 
enough  to  receive  twenty  dollars,  and  then  sold  him 
for  forty-three  dollars :  did  he  gain  or  lose  by  the 
bargain  1  and  how  much  ? 


SECTION 

A..  1.  What  cost  three  yards  of  tape,  at  two  cents 
a  yard  ?  * 

2.  What  cost  four  apples,  at  two  cents  apiece  ? 

*  The  pupil  should  be  made  to  observe,  that  three  yards  will  cost 
three  times  as  much  as  one  yard  ;  and  say,  If  one  yard  cost  two 
cents,  three  yards  will  cost  three  times  two  cents.  He  should  be 
made  to  give  this  reason  for  the  solution  of  Qfach  question,  varying 
the  number  according  to  the  question.  * 


32  ARITHMETIC.  [PART  1. 

3.  What  cost  five  peaches,  at  three  cents  apiece  ? 

4.  What  must  you  give  for  two  oranges,  at  six 
cents  apiece? 

5.  What  would  be  the  price  of  three  barrels  of 
cider,  at  three  dollars  a  barrel  ? 

6.  If   one   orange  is    worth    three    apples,    how 
many  apples  are  four  oranges  worth  ? 

7.  What  are  two  barrels  of  flour  worth,  at  five 
dollars  a  barrel  ? 

8.  What  cost  three  yards  of  cloth,  at-  four  dollars 
a  yard  ? 

9.  What    cost    two   pounds  of  raisins,  at  eight 
cents  a  pound  ? 

10.  What  cost  three  lemons,  at  six  cents  apiece  ? 

11.  If  a  man  travel  three  miles  in  an  hour,  how 
many  miles  will  he  travel  in  four  hours  ? 

12.  What  will  five  pairs  of  shoes  come  to,  at  two 
dollars  a  pair? 

13.  What  is  the  price  of  seven  yards  of  cloth,  at 
three  dollars  a  yard  ? 

14.  What  is  the  value  of  two  pounds  of  beef,  at 
seven  cents  a  pound  ? 

15.  If   there  are    three  feet    in    one  yard,  how 
many  feet  are  there  in  four  yards  ? 

16.  How  many  feet  are  there  in  seven  yards  ? 

17.  How  many  feet  are  there  in  six  yards  and  two 
feet? 

*  18.  If  a  man  earn  seven  dollars  in  one  week,  how 
much  would  he  earn  in  five  weeks  ? 

19.  What  cost  seven  hundred  weight  of  sugar, 
at  nine  dollars  a  hundred  weight  ? 

20.  What  cost  seven  pounds  of  sugar,  at  ten  cents 
a  pound  ? 

21.  If  one  half  yard  of  cloth  cost  three  dollars, 
what  would  three  yards  cost  ? 

22.  If  one  quarter   of  a    yard  of  cloth  cost  two 
dollars,  what  is  that  a  yard  ? 


SECT.  2.]  ARITHMETIC.  33 

23.  How  many  yards  of  cloth  are  there  in  seren 
pieces,  each  piece  containing  ten  yards  ? 

24.  What  will  five  barrels  of  flour  cost,    at  six 
dollars  a  barrel  1 

25.  If  a  man  can  travel  four  miles  in  an  hour, 
how  far  can  he  travel  in  eight  hours  ? 

26.  If  it  take  four  bushels  of  wheat  to  make  a 
barrel  of  flour,  how  many  bushels  will    it  tak;e  to 
make  seven  barrels  ? 

H.   1.  Two  times  one  are  how  many  ?* 

2.  Two  times  two  are  how  many  ? . 

3.  .Two  times  three  are  how  many  1 

4.  Two  times  four  are  how  many  ? 

5.  Two  times  five  are  how  many  ? 

6.  Two  times  six  are  how  many  ? 

7.  Two  times  seven  are  how  many  ? 

8.  Two  times  eight  are  how  many  ? 

9.  Two  times  nine  are  how  many  ? 

10.  Two  times  ten  are  how  many? 

11.  Three  times  one  are  how  many? 

12.  Three  times  two  are  how  many  ? 

13.  Three  times  three  are  how  many  ? 

14.  Three  times  four  are  how  many  ? 

15.  Three  times  five  are  how  many? 

16.  Three  times  six  are  how  many  ? 

17.  Three  times  seven  are  how  many  ? 

18.  Three  times  eight  are  how  many  ? 

19.  -Three  times  nine  are  how  many  ? 
£0.  Three  times  ten  are  how  many  ? 

21.  Four  times  one  are  how  many  ? 

22.  Four  times  two  are  how  many  ? 

23.  Four  times  three  are  how  many  ? 

24.  Four  times  four  are  how  many? 

25.  Four  times  five  are  how  many  ? 

26.  Four  times  six  are  how  many  ? 

*  See  the  Key. 


34  ARITHMETIC. 

27.  Four  times  seven  are  how  many  1 

28.  Four  times  eight  are  how  many  ? 

29.  Four  times  nine  are  how  many  ? 

30.  Four  times  ten  are  how  many  ? 

31.  Five  times  one  are  how  many? 

32.  Five  times  two  are  how  many  ? 

33.  Five  times  three  are  how  many? 

34.  Five  times  four  are  how  many  ? 

35.  Five  times  five  are  how  many  ? 

36.  Five  times  six  are  how  many  ? 

37.  Five  times  seven  are  how  many  ? 

38.  Five  times  eight  are  how  many  ? 

39.  Five  times  nine  are  how  many  ? 

40.  Five  times  ten  are  how  many? 

41.  Six  times  one  are  how  many  ? 

42.  Six  times  two  are  how  many  ? 

43.  Six  times  three  are  how  many  1 

44.  Six  times  four  are  how  many  ? 

45.  Six  times  five  are  how  many  ? 

46.  Six  times  six  are  how  many  ? 

47.  Six  times  seven  are  how  many  ? 

48.  Six  times  eight  are  how  many  ? 

49.  Six  times  nine  are  how  many  ? 

50.  Six  times  ten  are  how  many  ? 

51.  Seven  times  one  are  how  many? 

52.  Seven  times  two  are  how  many  ? 

53.  Seven  times  three  are  how  many  ? 

54.  Seven  times  four  are  how  many? 

55.  Seven  times  five  are  how  many  ? 

56.  Seven  times  six  are  how  many  ? 

57.  Seven  times  seven  are  how  many  ? 

58.  Seven  times  eight  are  how  many  ? 

59.  Seven  times  nine  are  how  many  ? 

60.  Seven  times  ten  are  how  many  ? 

61.  Eight  times  one  are  how  many? 

62.  Eight  times  two  are  how  many  ? 

63.  Eierht  times  three  are  how  many  ? 


SECT.  2.1  ARITHMETIC.  35 

64.  Eight  times  four  are  how  many  ? 

65.  Eight  times  five  are  how  many  ? 

66.  Eight  times  six  are  how  many? 

67.  Eight  times  seven  are  how  many  ? 

68.  Eight  times  eight  are  how  many  ? 

69.  Eight  times  nine  are  how  many  ? 

70.  Eight  times  ten  are  how  many  ? 

71.  Nine  times  one  are  how  many  ? 

72.  Nine  times  two  are  how  many  ? 

73.  Nine  times  three  are  how  many  ? 

74.  Nine  times  four  are  how  many  f 

75.  Nine  times  five  are  how  many  ? 

76.  Nine  times  six  are  how- many? 

77.  Nine  times  seven  are  how  many  ? 

78.  Nine  times  eight  are  how  many  ? 

79.  Nine  times  nine  are  how  many? 

80.  Nine  times  ten  are  how  many  ? 

81.  Ten  times  one  are  how  many? 

82.  Ten  times  two  are  how  many  ? 

83.  Ten  times  three  are  how  many  ? 

84.  Ten  times  four  are  how  many  ? 

85.  Ten  times cfive  are  how  many? 

86.  Ten  times  six  are  how  many  ? 

87.  Ten  times  seven  are  how  many  ? 

88.  Ten  times  eight  are  how  many  ? 

89.  Ten  times  nine  are  how  many  ? 

90.  Ten  times  ten  are  how  'many  ? 

CJ.   1.  Two  times  two  are  how  many  times  one? 

2.  Three  times  two  are  how  many  times  one  ? 

3.  Four  times  two  are  how  many  times  one  ? 

4.  Five  times  two  are  how  many  times  one  ? 

5.  Seven  times  two  are  how  many  ? 

6.  Nine  times  two  are  how  many? 

7.  Six  times  two  are  how  many  ? 

8.  Eight  times  two  are  how  many  ? 

9.  Ten  times  two  are  how  many  ? 


36  ARITHMETIC.  IP*RT  1- 

10.  Two  times  three  are  how  many  ? 

11.  Three  times  three  are  how  many? 

12.  Four  times  three  are  how  many? 

13.  Five  times  three  are  how  many? 
„    14.  Six  times  three  are  how  many  ? 

15.  Eight  times  three  are  how  many  ? 

16.  Seven  times  three  are  how  many  ? 

17.  Ten  times  three  are  how  many  ? 

18.  Nine  times  three  are  how  many  ? 

19.  Two  times  four  are  how  many? 

20.  Six  times  four  are  how  many  ? 

21.  Four  times  four  are  how  many? 

22.  Seven  times  four  are  how  many  ? 

23.  Nine  times  four  are  how  many  ? 

24.  Three  times  four  are  how  many? 

25.  Five  times  four  are  how  many  ? 

26.  Ten  times  four  are  how  many  ? 

27.  Eight  times  four  are  how  many  ? 

28.  Two  times  five  are  how  many  ? 

29.  Five  times  five  are  how  many  ? 

30.  Three  times  five  are  how  many  ? 

31.  Six  times  five  are  how  many? 

32.  Two  times  six  are  how  many  ? 

33.  Four  times  five  are  how  many  ? 

34.  Seven  times  five  are  how  many  ? 

35.  Three  times  six  are  how  many? 

36.  Seven  times  six  are  how  many  ? 

37.  Seven  times  seven  are  how  many  ? 

38.  Four  times  eight  are  how  many  ? 

39.  Six  times  seven  are  how  many  ? 

40.  Eight  times  nine  are  how  many  ? 

41.  Six  times  eight  are  how  many? 

42.  Three  times  seven  are  how  many  ? 

43.  Four  times  nine  are  how  many  ? 

44.  Three  times  eight  are  how  many  ? 

45.  Six  times  six  are  how  many  ? 

46.  Six  times  nine  are  how  many  ? 


SECT.  2.]  ARITHMETIC.  37 

47.  Nine  times  five  are  how  many  ? 

48.  Four  times  six  are  how  many  ? 

49.  Two  times  nine  are  how  many  ? 

50.  Seven  times  nine  are  how  many? 

51.  Nine  times  eight  are  how  many? 

52.  Two  times  eight  are  how  many  ? 

53.  Three  times  ten  are  how  many  ? 

54.  Eight  times  seven  are  how  many  ? 

55.  Five  times  six  are  how  many? 

56.  Five  times  eight  are  how  many  ? 

57.  Two  times  seven  are  how  many  ? 

58.  Two  times  six  are  how  many  ? 

59.  Eight  times  six  are  how  many  ? 

60.  Four  times  seven  are  how  many  ? 

61.  Eight  times  eight  are  how  many  ? 

62.  Ten  times  five  are  how  many  ? 

63.  Seven  times  ten  are  how  many  ? 

64.  Ten  times  ten  are  how  many  ? 

65.  Nine  times  six  are  how  many  1 

66.  Five  times  nine  are  how  many  ? 

67.  Three  times  nine  are  how  many  ? 

68.  Nine  times  seven  are  how  many  ? 

69.  Five  times  ten  are  how  many  ? 

70.  Seven  times  eight  are  how  many  ? 

71.  Five  times  seven  are  how  many  ? 

72.  Ten  times  eight  are  how  many  ? 

73.  Ten  times  seven  are  how  many  ? 

74.  Nine  times  ten  are  how  many  ? 

75.  Eight  times  five  are  how  many  ? 

76.  Nine  times  nine  are  how  many  ? 

77.  Four  times  ten  are  how  many  ? 

78.  Ten  times  six  are  how  many  ? 

79.  Eight  times  ten  are  how  many  ? 

80.  Ten  times  nine  are  how  many  ? 

JD.   1.  What   cost  three  yards  of  cloth,  at  five 
dollars  a  yard  ? 
4 


38  ARITHMETIC.  [PART  1. 

2.  What  cost  four  oranges,  at  six  cents  apiece  ? 

3.  What   cost   seven    barrels  of  cider,  at  three 
dollars  a  barrel  1 

4.  How  much  do  three  barrels  of  beer  come  to, 
at  seven  dollars  a  barrel  ? 

5.  What    cost    four     firkins   of  butter,  at   eight 
dollars  a  firkin? 

6.  What  do  nine  pounds  of  veal  come  to,  at  six 
cents  a  pound  ? 

7.  What  cost  six  reams  of  paper,  at  five  dollars 
per  ream  ? 

8.  What  cost  eight  pairs  of  shoes,  at  three  dollars 
a  pair  ? 

"9.  What  is  the  value  of  nine  yards  of  cloth,  at 
six  dollars  a  yard  ? 

10.  If  a  man  travel  five  miles  in  an  hour,  how 
many  miles  will  he  travel  in  nine  hours? 

11.  There  is  an  orchard  consisting  often  rows  of 
trees,  and  nine  trees  in  each  row ;  how  many  trees 
are  there  in  the  orchard  ? 

12.  On  a  chess-board  there  are  eight   rows  of 
squares,  and  eight  squares  in  each  row ;  how  many 
squares  are  there  on  the  board  ? 

13.  In  one  penny  there  are  four  farthings ;  how 
many  farthings  are  there  in  six  pence  ? 

14.  How    many    farthings   are    there   in    eight 
pence  ? 

15.  How    many    farthings    are    there    in    nine 
pence  ? 

16.  How  many  farthings  are  there  in  ten  pence  ? 

17.  In  one  shilling  there  are  twelve  pence  ;   how 
many  farthings  are  there  in  a  shilling  ? 

18.  In  one  pint  there  are   four  gills ;  how  many 
gills  are  there  in  five  pints  ? 

19.  In  one  quart  there  are  two  pints  ;  how  many 
pints  are  there  in  six  quarts  ? 

20.  How  many  pints  are  there  in  three  quarts  ? 


SECT.  2.]  ARITHMETIC.  39 

21.  How  many  gills  are  there  in  six  pints  ? 

22.  How  many  gills  are  there  in  one  quart  ? 

23.  How  many  gills  are  there  in  three  quarts  ? 

24.  In  one  gallon  there  are    four  quarts;    how 
many  quarts  are  there  in  three  gallons  1 

25.  How  many  quarts  are  there  in  five  gallons  ? 

26.  How  many  quarts  are  there  in  seven  gallons  ? 

27.  How  many  pints  are  there  in  one  gallon  1 

28.  How  many  pints  are  there  in  three  gallons  ? 

29.  How  many  gills  are  there  in  one  gallon  ? 

30.  How  many  gills  are  there  in  five  quarts  ? 

31.  How  many  gills  are  there  in  two  gallons  ? 

32.  A  person  bought  two  oranges,  at  six  cents 
apiece ;  and  seven  lemons,  at  four  cents  apiece ;  and 
five  pears,  at  two  cents  apiece  :  how  much  did  the 
whole  come  to  ? 

33.  If  one  pint  of  gin  cost  eight  cents,  what  will 
one  quart  cost  ? 

34.  If  one  gill  of  brandy  cost  four  cents,  what 
will  one  quart  cost  ? 

35.  If  one  gill  of  beer  cost  two  cents,  what  will 
one  gallon  cost  ? 

36.  If  a  stage  runs  seven  miles  in  an  hour,  how 
far  will  it  run  in  nine  hours  ? 

37.  Two    men   start  from   the  same  place,   and 
travel  different  ways;  one  travels  two  miles  in  an 
hour  ;    the  other  travels  three  miles   in    an  hour  : 
how  far  apart  will  they  be  at  the  end  of  one  hour  '? 
How  far  at  the  end  of  two  hours  ?  How  far  at  the 
end  of  three  hours  ?     How  far  at  the  end  of  four 
hours  ] 

38.  Two    men    start  from  the  same  place,  and 
travel  the  same  way ;  one  travels  at  the  rate  of  two 
miles    in  an  hour,  the  other  four  :    how  far  apart 
will  they  be  in  one  hour  ?  How  far  in  two  hours  ? 
How  far  in  four  hours  1 

39.  If  three  men  can  do  a  piece  of  work  in  two 


40  ARITHMETIC.  [PART  1. 

days,    how  many  days  would  it  take  one 'man  to 
do  it? 

40.  If  four  men  can  do  a  piece  of  work  in  ^five 
days,  how  many  days  would  it  take  one  man  to  do  it  ? 

41.  If  six  men  can  do  a  piece  of  work  in  seven 
days,  how  many  men  would  it  take  to  do  it  in  one 
day? 

42.  If  a  quantity  of  provisions  will  serve  three 
men  five  days,  how  many  men  would  it  serve  one 
day? 

43.  If  a  quantity  of  provisions  will   serve   five 
men  seven  days,  how  many  days  would  it  serve  one 
man  ? 

44.  If  fifteen  dollars'  worth  of  provisions  will  serve 
eight  men  five  days,  how  many  days  will  it    serve 
one  man  ? 

45.  A  man  had  a  piece  of  work  to  perform  which 
seven  men  could  do  in  nine  days,  but  it  was  neces- 
sary that   the  whole    should   be  completed  in  one 
day  ;  how  many  men  must  he  employ  ? 

46.  If  the  interest  of  one  dollar  is  six  cents  a  year, 
what  would  be  the  interest  of  ten  dollars  for  the 
same  time  ? 

47.  If  the  interest  of  one  dollar,  is  six  cents  for 
one  year,  what  would  be  the  interest  of  it  for  two 
years  ?  For  three  years  ?  For  seven  years  ? 

48.  If  a  man  can  earn  seven  shillings  in  a  day, 
how  many  shillings  will  he  earn  in  six  days  ? 

49.  If  a  man  can  earn  eight  dollars  in  a  month, 
how  much  can  he  earn  in  six  months  ? 

50.  At  five  dollars  a  week,  what  will  nine  weeks' 
board  come  to  ? 

51.  A  lady  bought  three  yards  of  cambric  at  two 
dollars  a  yard,  seven  yards  of  silk  for  three  dollars 
a  yard,  five  yards  of  riband  for  four  dollars,  and 
some  crape  for  two  dollars ;    she  paid  four  ten-dollar 
bills  ;    how  much  must  she  receive  back  again  ? 


SECT.  3.]  ARITHMETIC.  41 


SECTION    III. 

A.   1.  How  many  apples,  at  one  cent  apiece,  can 
you  buy  for  four  cents  ? 

2.  How  many  pears,  at  two  cents  apiece,  can  you 
buy  for  four  cents  ? 

3.  How  many  peaches,  at  three  cents  apiece,  can 
you  buy  for  six  cents  ? 

4.  How  many  apples,  at  two  cents  apiece,  can 
you  buy  for  six  cents  ?     How  many  for  eight  cents  1 

5.  How  many  pears,   at  three  cents  apiece,  can 
you  buy  for    nine    cents  ?     How  many  for  twelve 
cents  ? 

6.  If  you  have  eight  apples  to  give  to  four  boys, 
how  many  can  you  give  to  each  ? 

7.  If  a  man  travel  six  miles  in  two  hours,  how 
many  miles  does  he  travel  in  an  hour  ? 

8.  If  a  man  travel  three  miles  in   an  hour,  how 
many  hours  will  it  take  him  to  travel  nine  miles  1 

9.  How  many  yards  of  cloth,   at  three  dollars  a 
yard,  can  you  buy  for  fifteen  dollars? 

10.  If  you  had  sixteen  cents,  how  many  cakes 
could  you  buy  at  four  cents  apiece  ? 

11.  If  you  had  ten  dollars,  how  much  cloth  could 
you  buy  at  five  dollars  a  yard  ? 

12.  If  you  had  twelve  apples  to  give  to  six  of 
your  companions,  how  many  would  you  give  them 
apiece  1 

13.  If  a  man  can  travel  six  miles  in  an  hour,  how 
long  will  it  take  him  to  travel  eighteen  miles  ? 

14.  If  a  man  can  travel  five  miles  in  an  hour,  how 
long  will  it  take  him  to  travel  twenty  miles  ? 

15.  In  a  certain  orchard  there  are  twenty-four 
trees  standing  in  rows,  and  there  are  three  trees  in 
each  row  ;  how  many  rows  are  there  ? 

4* 


42  ARITHMETIC.  [PART  1. 

16.  In  an  orcnard  there  are  twenty-one  trees,  and 
there  are  seven  trees  in  each  row ;  how  many  rows 
are  there  ? 

17.  A    man  paid  twenty-seven  dollars  for  some 
sheep,  and  he  gave  nine  dollars    apiece  for  them ; 
how  many  sheep  did  he  buy  ? 

18.  A  man  paid  twenty-eight  dollars    for    seven 
barrels  of  cider  ;  how  much  did  he  give  a  barrel  ? 

19.  At  five  cents  apiece,  how  many  oranges  can 
you  buy  for  thirty  cents  ? 

20.  Twenty-five  are  how  many  times  five  1 

21.  Thirty-two  are  how  many  times  four?    How 
many  times  eight  ? 

22.  Thirty-five  are  how  many  times  seven  ?  How 
many  times  five  ? 

23.  Thirty-six  are  how  many  times  six?     How 
many  times  nine  ?    How  many  times  four  ? 

B.  Remark.  When  any  thing,  or  any  number,  is 
divided  into  two  equal  parts,  one  of  the  parts  is 
called  the  half  of  the  thing  or  the  number. 

1.  If  an  apple  is  worth  two  cents,  what  is  one 
half  of  it  worth  ? 

2.  What  is  one  half  of  two  cents  ? 
Ans.  One  cent. 

Q.  Why? 

A.  Because,  if  you  divide  two  cents  into  two 
equal  parts,  one  of  the  parts  is  one  cent. 

3.  If  you    can  buy  a  cake  for  two  cents,  how 
much  can  you  buy  for  one  cent  ? 

4.  One  is  what  part  of  two? 
Ans.  One  is  the  half  part  of  two. 

5.  Two  times  one  are  how  many  times  two  ? 

6.  If  you  can  buy  one  pear  for  two  cents,  how 
many  can  you  buy  for  three  cents  ? 

7.  Three  are  how  many  times  two  ? 
Ans.  Once  two  and  one  half  of  two. 


SECT.  3.]  ARITHMETIC.  43 

8.  Four  are  how  many  times  two  1 

9.  If  two  shillings  will    buy  one   yard  of  cotton 
cloth,  how  many  yards  will  five  shillings  buy  1 

10.  Five  are  how  many  times  two? 
Ans.  Two  times  two  and  half  of  two. 

11.  Six  are  how  many  times  two  ? 

12.  If  two  dollars  will  buy  a  yard  of  cloth,  how 
many  yards  will  seven  dollars  buy  ? 

13.  How  many  halves  make  a  whole  one? 

14.  Eight  are  how  many  times  two  ? 

15.  Nine  are  how  many  times  two  I 

16.  Ten  are  how  many  times  two  1 

Remark.  When  any  thing,  or  any  number,  is  di- 
vided into  three  equal  parts,  one  of  those  parts  is 
called  the  third  part  of  the  thing  or  number.  When 
it  is  divided  into  four  equal  parts,  one  part  is  called 
the  fourth  part;  and  so  on. 

17.  If  a  yard  of  cloth  be  worth  three  dollars,  and 
it  be  cut  into  three  equal  pieces,  what  will  one  of 
the  pieces  be  worth  1  that  is,  what  will  one  third  of 
a  yard  be  worth  1 

18.  What  is  a  third  of  three? 

19.  Suppose  the  yard  of  cloth  to  be  cut  as  before, 
what  will  two  pieces  of  it  cost  ?  that  is,  what  will 
two  thirds  of  a  yard  cost  ? 

20.  What  is  two  thirds  of  three  ? 

21.  If  three  shillings  will  buy  one  bushel  of  corn, 
what  part  of  a  bushel  will  one  shilling  buy  ?     What 
part  of  a  bushel  will  two  shillings  buy  ? 

22.  One  is  what  part  of  three  ? 

Ans.  One  is  the  third  part  of  three;  or  one  third 
of  three. 

23.  Two  is  what  part  of  three  ? 

Ans.  Two  is  two  times  the  third  part  of  three 
or  two  thirds  of  three.* 

*  The  instructor  may  use  either  of  these  expressions  ;  the  former 
will  be  most  easily  understood  by  the  pupil  5  it  would  be  well;  there- 


44  ARITHMETIC.  [PART  1. 

24.  Three  times  one  are  how  many  times  three  ? 

25.  If  you  can  buy  a  barrel  of  cider  for  three 
dollars,  how  much  can  you  buy  for  four  dollars? 
How  much  for  five  dollars  ? 

26.  How  many  thirds  make  a  whole  one  ? 

27.  Five  are  how  many  times  three? 
Ans.  Once  three,  and  two  thirds  of  three. 

28.  Six  are  how  many  times  three  ? 

29.  If  you  can  buy  a  barrel  of  fish  for  three  dol- 
lars,  how   much  can  you   buy  for  seven  dollars? 
How  much  for  eight  dollars? 

30.  What  do  you  understand  by  a  third,  and  by 
two  thirds  of  any  thing  ? 

For  the  answer,  see  remark  after  example  16th. 

31.  Eight  are  how  many  times  three? 

32.  Nine  are  how  many  times  three? 

33.  Ten  are  how  many  times  three  ? 

34.  Eleven  are  how  many  times  three? 

35.  Twelve  are  how  many  times  three? 

36.  If  a  yard  of  cloth  be  worth  four  dollars,  and 
it  be  cut  into  four  equal  parts,  what  will  one  of  the 
parts  be  worth  ?  that  is,  what  is  one   fourth  of  it 
worth  ?     What  are  two  fourths  of  it  worth  ?     What 
are  three  fourths  of  it  worth  ? 

37.  If  you  can  buy  a  barrel  of  cider  for  four  dol- 
lars, how  much  can  you  buy  for  one  dollar  ?     How 
much  for  two  dollars  ?  How  much  for  three  dollars  ? 

38.  What  part  of  four  is  one  ? 
Ans.  One  is  the  fourth  part  of  four. 

39.  What  part  of  four  is  two  ? 
Ans.  Two  fourths  of  four. 

40.  What  part  of  four  is  three  ? 
Ans.  Three  fourths  of  four. 

41.  How  many  fourths  make  a  whole  one? 

42.  If  you  can  buy  a  bushel  of  corn  for  four  shil- 

fore,  for  the  instructor  to  use  the  former  frequently,  though  the  latter 
»  used  for  the  most  part  in  this  treatise. 


SECT.  3.]  ARITHMETIC.  45 

lings,  how  much  can  you  buy  for  five  shillings  ? 
How  much  for  six  shillings  1  How  much  for  seven 
shillings  ? 

43.  Five  are  how  many  times  four  ? 
Ans.  Once  four,  and  one  fourth  of  four. 

44.  Six  are  how  many  times  four  1 
Ans.  Once  four,  and  two  fourths  of  four. 

45.  Seven  are  how  many  times  four  ? 
Ans.  Once  four,  and  three  fourths  of  four. 

46.  Eight  are  how  many  times  four? 

47.  If  four  bushels  of  corn  will  buy  one  yard  of 
cloth,  how  many  yards  will  nine  bushels  buy  ?  How 
many  yards  will  ten  bushels  buy  ?  How  many  yards 
will  eleven  bushels  buy  ? 

48.  What  do  you  understand  by  one  fourth,  two 
fourths,  or  three  fourths  of  any  thing  1 

See  remark  after  example  16th. 

49.  Ten  are  how  many  times  four  ? 

50.  Eleven  are  how  many  times  four  1 

51.  Twelve  are  how  many  times  four? 

52.  Thirteen  are  how  many  times  four  ? 

53.  Fourteen  are  how  many  times  four  ? 

54.  Fifteen  are  how  many  times  four  ? 

55.  Sixteen  are  how  many  times  four? 

56.  If  a  barrel  of  flour  be  worth  five  dollars,  and 
it  be  divided  equally  among  five  men,  what  will  one 
man's  share  be  worth  1  that  is,  what  is  one  fifth  of 
a  barrel  worth  ?     What  are  two  fifths  of  it  worth  ? 
What  are  three  fifths  of  it  worth  ?     What  are  four 
fifths  of  it  worth  ? 

57.  If  five  dollars  will   buy  one  box  of  butter, 
what  part  of  a  box  will  one  dollar  buy  ?     What  part 
will  two  dollars  buy  ?     What  part  will  three  dollars 
buy  ?     What  part  will  four  dollars  buy  ? 

58.  What  part  of  five  is  one  ? 
Ans.  One  is  the  fifth  part  of  five. 

59.  Two  is  what  part  of  five  ? 


46  ARITHMETIC.  [PART  1 

Arts.  Two  fifths  of  five. 

60.  Three  is  what  part  of  five  ? 
-4ns.  Three  fifths  of  five. 

61.  Four  is  what  part  of  five  ? 

62.  How  many  fifths  make  a  whole  one  ? 

63.  If  cherries  are  five  cents  a  quart,  how  many 
quarts  can  you  buy  for  six  cents  ?     How  many  for 
seven  cents  1     How  many  for  eight  cents  ?     How 
many  for  nine  cents  ?    How  many  for  eleven  cents  ? 
How  many  for  thirteen  cents  ? 

64.  What  do  you  understand  by  one  fifth,  two 
fifths,  &/c.  of  any  thing  ? 

See  remark  after  example  16th. 

65.  Seven  are  how  many  times  five  ? 
Ans.  Once  five  and  two  fifths  of  five. 

66.  Eight  are  how  many  times  five  ? 

67.  Nine  are  how  many  times  five  ? 

68.  Ten  are  how  many  times  five  ? 

69.  Eleven  are  how  many  times  five  ? 

70.  Twelve  are  how  many  times  five  ? 

71.  Thirteen  are  how  many  times  five  ? 

72.  Fourteen  are  how  many  times  five  ? 

73.  Fifteen  are  how  many  times  five  ? 

74.  If  a  barrel  of  beef  cost  six  dollars,  and  it 
were  divided  into  six  equal  parts,  what  would  one  of 
the  parts  be  worth  1  that  is,  what  is  one  sixth  of  it 
worth?     What  are  two  sixths  of  it  worth?     What 
are  three  sixths  of  it  worth  ?     Four  sixths  ?     Five 
sixths  ? 

75.  If  fish  is  worth  six  dollars  a  barrel,  what  part 
of  a  barrel  will  one  dollar  buy  ?   What  part  of  a  bar- 
rel will  two  dollars  buy  ?    Three  dollars  ?    Four  dol- 
"ars?  Five  dollars? 

76.  What  part  of  six  is  one  ? 
Ans.  One  is  the  sixth  part  of  six. 

77.  What  part  of  six  is  two  ? 
Ans.  Two  sixths  of  six. 


SECT.  3.]  ARITHMETIC.  47 

78.  Three  is  what  part  of  six  ? 
Ans.  Three  sixths  of  six. 

79.  Four  is  what  part  of  six  ? 

80.  How  many  sixths  make  a  whole  one  ? 

81.  How  much  rye,  at  six  shillings^  bushel,  can 
you  buy  for  seven  shillings  ?     How  much  for  eight 
shillings?    Nine  shillings?    Ten  shillings?    Eleven 
shillings?    Twelve   shillings?    Thirteen   shillings? 
Fifteen  shillings?     Seventeen  shillings? 

82.  What  do  you  understand  by  one  sixth,  two 
sixths,  &c.  ? 

83.  Eight  are  how  many  times  six? 
Ans.  One  time  six  and  two  sixths  of  six. 

84.  Nine  are  how  many  times  six  ? 

85.  Ten  are  how  many  times  six? 

86.  Eleven  are  how  many  times  six  ? 

87.  Twelve  are  how  many  times  six? 

88.  Thirteen  are  how  many  times  six? 

89.  Fourteen  are  how  many  times  six? 

90.  Fifteen  are  how  many  times  six? 

91.  If  coal  is  worth  seven  dollars  a  chaldron,  what 
is  one  seventh  of  a  chaldron  worth  ?     What  are  two 
sevenths  of  a  chaldron  worth  ?      Three  sevenths  ? 
Four  sevenths  ?    Five  sevenths  ?     Six  sevenths  ? 

92.  At   the   rate  of  seven  dollars  a   yard,  how 
much  broadcloth  can  you  buy  for  one  dollar  ?    How 
much  for  two  dollars  ?    How  much  for  three  dollars  ? 
How  much  for  four  dollars?    How  much  for  five  dol- 
lars ?     How  much  for  six  dollars  ?     How  much  for 
eight  dollars  ?     How  much  for  ten  dollars  ?     How 
much  for  twelve  dollars  ?      How  much  for  fifteen 
dollars  ? 

93.  What  part  of  seven  is  one? 
Ans.  One  is  one  seventh  of  seven. 

94.  What  part  of  seven  is  two? 
Ans.  Two  sevenths  of  seven. 

95.  What  part  of  seven  is  three  ? 


48  ARITHMETIC.  [PART  1. 

.  96.  Four  is  what  part  of  seven  ? 

97.  Five  is  what  part  of  seven  ? 

98.  What  do  you  understand  by  one  seventh,  two 
sevenths,  &c.  of  any  thing  ? 

99.  How  many  sevenths  make  a  wnole  one  ? 

100.  Nine  are  how  many  times  seven  ? 

101.  Ten  are  how  many  times  seven? 

102.  Eleven  are  how  many  times  seven? 

103.  Twelve  are  how  many  times  seven  ? 

104.  Thirteen  are  how  many  times  seven? 

105.  Fourteen  are  how  many  times  seven  ? 

106.  Fifteen  are  how  many  times  seven? 

107.  Sixteen  are  how  many  times  seven  ? 

108.  When  wheat   is   eight   shillings    a   bushel, 
what  is  one  eighth  of  a  bushel  worth  ?     What  are 
two  eighths  of  a  bushel  worth?     What  are  three 
eighths  of  a  bushel  worth  ?     What  are  four  eighths 
of  a  bushel  worth  ?     Five  eighths  ?     Six  eighths  ? 
Seven  eighths  ? 

109.  When  wood  is  eight   dollars  a  cord,  what 
part  of  a  cord  can  you  buy  for  a  dollar  ?    What  part 
of  a  cord  can  you  buy  for  two  dollars  ?     What  part 
for  three  dollars  ?  What  part  for  four  dollars  ?  What 
part  for  five  dollars?      What  part  for  six  dollars? 
What  part  for  seven  dollars?     How  much  can  you 
buy  for  nine  dollars  ?     How  much  for  ten   dollars  ? 
How  much  for  eleven  dollars  ?     How  much  for  thir- 
teen dollars  ?     How  much  for  fifteen  dollars  ?    How 
much  for  nineteen  dollars  ? 

1 10.  What  part  of  eight  is  one  ? 

111.  What  part  of  eight  is  two? 

112.  Three  is  what  part  of  eight? 

113.  Four  is  what  part  of  eight? 

114.  Five  is  what  part  of  eight? 

115.  What  do  you  understand  by  one  eighth,  two 
eighths,  &/c.  of  any  number  ? 

116.  Seven  is  what  part  of  eight? 


SECT.  3.]  ARITHMETIC.  49 

117.  How  many  eighths  make  a  whole  one  ? 

118.  Ten  are  how  many  times  eight? 

119.  Eleven  are  how  many  times  eight? 

120.  Twelve  are  how  many  times  eight? 

121.  Thirteen  are  how  many  times  eight? 

122.  Fourteen  are  how  many  times  eight? 

123.  When  sugar  is  nine  dollars  a  hundred  weight, 
what  is  one  ninth  of  a  hundred  weight  worth  ?  What 
are  two  ninths  of  a  hundred  weight  worth  ?     Three 
ninths  ?     Four  ninths  ?     Five  ninths  ?     Six  ninths  ? 
Seven  ninths  ?  Eight  ninths  ? 

124.  When  rye  is  nine  shillings  a  bushel,  what 
part  of  a  bushel  can  you  buy  for  one  shilling  ?  What 
part  for  two  shillings  ?     What  part  for  three  shil- 
lings ?  For  four  shillings  ?  For  five  shillings  ?  For 
six  shillings?      Seven  shillings?     Eight  shillings? 
How  much  for  ten  shillings  ?  For  thirteen  shillings  ? 
For  fourteen  shillings  ?    Sixteen  shillings  ?    Twenty 
shillings  ? 

125.  What  do  you  understand  by  one  ninth,  two 
ninths,  three  ninths,  &c.,  of  any  number? 

126.  Three  is  what  part  of  nine  ? 

127.  Four  is  what  part  of  nine  ? 

128.  Five  is  what  part  of  nine  ? 

129.  Seven  is  what  part  of  nine  ? 

130.  How  many  ninths  make  a  whole  one  ? 

131.  Thirteen  are  how  many  times  nine  ? 

132.  Fifteen  are  how  many  times  nine  ? 

133.  Seventeen  are  how  many  times  nine  ? 

134.  When  hay  is  ten  dollars  a  ton,  what  is  one 
tenth  of  a  ton  worth  ?     What  are  two  tenths  of  a 
ton  worth  ?    What  are  three  tenths  of  a  ton  worth  ? 
Four  tenths?     Five  tenths?     Six  tenths?     Seven 
tenths  ?  Eight  tenths  ?  Nine  tenths  ? 

135.  When  sugar  is  ten  dollars  a  hundred  weight, 
what  part  of  a  hundred  weight  can  you  buy  for  one 
dollar  ?  What  part  for  two  dollars  ?  What  part  for 


50  ARITHMETIC.  [PART  1 

three  dollars  ?  What  part  for  four  dollars  ?  What 
part  for  five  dollars'?  Six  dollars?  Seven  dollars? 
Eight  dollars?  Nine  dollars?  How  much  can  you 
buy  for  eleven  dollars  ?  Thirteen  dollars  ?  Fifteen 
dollars?  Seventeen  dollars ? 

136.  What  do  you  understand  by  one  tenth,  two 
tenths,  three  tenths,  &c.,  of  any  thing?* 

137.  How  many  tenths  make  a  whole  one  ?  * 

C.  Instead  of  writing  the  names  of  numbers,  it 
is  usual  to  express  them  by  particular  characters, 
called  jigures. 

One  is  written 1 

Two  is  written 2 

Three  is  written     3 

Four  is  written 4 

Five  is  written 5 

Six  is  written 6 

Seven  is  written 7 

Eight  is  written 8 

Nine  is  written 9 

Ten  is  written 10 

1.  Eleven  times  one  are  how  many  times  2  ? 

2.  Twelve  are  how  many  times  2  ?    3  ?    4  ? 

3.  Fourteen  are  how  many  times  2  ?    4  ?    3  ? 

4.  If  you  had  fifteen  cents,  how  many  cakes  could 
you  buy  at  4  cents  apiece  ?     How  many  at  2  cents 
apiece  ?  How  many  at  3  cents  apiece  ?    How  many 
at  5  cents  apiece  ? 

5.  Fifteen  are  how  many  times  4  ?    2  ?    3  ?    5  ? 

6.  Sixteen  are  how  many  times  5?    3?    6?    2? 
7?    4? 

7.  Seventeen  are  how  many  times  6?    2?  7?  3? 
5?    4? 

8.  Eighteen  are  how  many  times  4  ?    7  ?   9  ?    6 1 
3?   2?   5?   8? 

*  These  questions  should  frequently  be  put  to  the  learners. 


SECT.  3.]  ARITHMETIC.  51 

9.  Nineteen  are  how  many  times  3?   7?  4?  5? 
8?    6?    9?   2?    10? 

10.  Twenty  are  how  many  times  6?    2?    8?    3? 
9?   4?    10?   5?    7? 

11.  Twenty-one  are  how  many  times  7?    3?    8? 
2?   4?   6?   9?    5?    10? 

12.  Twenty-two  are  how  many  times  3?    8?    5? 
4?    9?    6?    7?    10?    2? 

13.  If  you  had  twenty-seven   dollars,  how  much 
cloth  could  you  buy  at  9  dollars  a  yard  ?    How  much 
at  6  dollars  a  yard?    How  much  at  4  dollars  a  yard? 
How  much  at  3  dollars  a  yard  ?     How  much  at  7 
dollars  a  yard?  How  much  at  8  dollars  a  yard?  How 
much  at  5  dollars  a  yard  ?    How  much  at  10  dollars 
a  yard  ? 

14.  Twenty-seven  are  how  many  times  9  ?    6  ?    4  ? 
3?    7?   8?    5?    10? 

15.  Twenty-four  are  how  many  times  6  ?    8  ?    7  ? 
5?   2?    10?   3?   4?   9? 

16.  Twenty-nine  are  how  many  times  3  ?    7  ?    5  ? 
9?   6?   8?   4?    10? 

17.  Twenty-three  are  how  many  times  4  ?    2  ?  7  ? 
8?   3?    9?    6?    5?    10?       ! 

18.  Twenty-five  are  how  many  times  3  ?    7  ?    2 1 
6?   9?    4?   8?    5?    10? 

19.  Thirty  are  how  many  times  10?    2?    3?    7? 
9?    6?    5?    4?    8? 

20.  Thirty-three  are  how  many  times  6?    8?    7? 
4?-  9?   5?    10?    3? 

21.  Twenty-six  are  how  many  times  9?  4?  7?  3? 
8?   5?    6?    10? 

22.  Thirty-five  are  how  many  times  5?    6?    3? 
7?   9?    10?   4?   8? 

23.  Thirty-eight  are  how  many  times  8  ?    6?    3? 
9?   5?   4?    7?    10? 

24.  Thirty-four  are  how  many  times  7?    3?    9? 
10!   6?   8?   4?   5? 


53  ARITHMETIC.  [PART  1. 

25.  Thirty-six  are  how  many  times  8 1   9  ?   4  ?    57 
3?   6?    7?    10? 

26.  Forty  are  how  many  times  8  ?  10?   6?  4?  3? 
9?   5?    7? 

27.  For  forty-seven  cents,  how  many  pounds  of 
meat  can  be  bought  at  6  cents  a  pound?   How  many 
pounds  at  8  cents?    How  many  at  9  cents?    How 
many  at  3  cents ?  How  many  at  5  cents?  How  many 
at  4  cents  ?    How  many  at  7  cents  ?    How  many  at  10 
cents  a  pound  ? 

28.  Forty-seven  are  how  many  times  6?    8?    9? 
3?   5?   4?    7?    10? 

29.  Forty-three  are  how  many  times  9?    8?    7? 
6?   4?   3?   5?    10? 

30.  Forty-five  are  how  many  times  10?  8?   3?  6? 
4?    7?   5?   9? 

31.  Forty-nine  are  how  many  times  6?    10?    5? 
9?   4?   8?   7? 

32.  Fifty-three  are  how  many  times  8 ?  5?  6?  4? 
7?   9?    10? 

33.  Fifty-seven  are  how  many  times  9  ?    7?    10? 
6?    5?    8?    4? 

34.  Fifty-five  are  how  many  times  6?  4?  8?  10? 
9?   7?   5? 

35.  Forty-eight  are  how  many  times  7  ?    5  ?    9  ? 
4?    6?    8?   10? 

36.  Fifty-four  are  how  many  times  5  ?    9  ?    6  ?    4  ? 
7?    10?    8? 

37.  Forty-four  are  how  many  times  4?  6?  9?  71 
5?   8?    10? 

38.  Fifty-eight  are  how  many  times  7?  6?  8?  4? 
9?   5?    10? 

39.  Forty-six  are  how  many  times  8?  10?  4?  6? 
9?   7?   5? 

40.  Fifty  are  how  many  times  9?  5?  4?   10?  8? 
6?   7? 


SECT.  3.]  ARITHMETIC.  53 

41.  Fifty-nine  are  how  many  times  4  ?  8?   7  ?  6? 
10?   9?   5? 

42.  Sixty-four  are  how  many  times  77578?  10? 
6?   9? 

43.  Sixty-eight  are  how  many  times  6 ?  8?  9?  7? 
10?   5? 

44.  Fifty-two  are  how  many  times  4  ?   6?   8?    10? 
5?   7?   9? 

45.  Sixty-three  are  how  many  times  5?    4?    6? 
10?   9?    7?   8? 

46.  Sixty-two  are  how  many  times  4?   10?  9?  7? 
8?    5?   6? 

47.  Seventy-three  are  how  many  times  10?   7? 
8?   6?   5?   9? 

48.  Seventy-five  are  how  many  times  7?   8?    10? 
5?    6?   9? 

49.  If  you   had   sixty-seven   dollars,   how   many 
barrels  of  flour  could  you  buy  at  5  dollars  a  barrel  ? 
How  many  at  7  dollars  a  barrel  ?     How  many  at  6 
dollars  a  barrel  ?     How  many  at  8  dollars  a  barrel  ? 
How  many  at  10  dollars  a  barrel  ?     How  many  at  9 
dollars  a  barrel  ? 

50.  Sixty-seven  are  how  many  times  5?    7?    6? 
8?    10?   9? 

51.  Seventy-four  are  how  many  times  10  ?  7?   8? 
6?   5?   9? 

52.  Sixty  are  how  many  times  9 ?    10?   6?  4?  7? 
5?   8? 

53.  Seventy-two  are  how  many  times  5  ?    7  ?    6  ? 
8?    10?    9? 

54.  Sixty-five  are  how  many  times  5  ?    10  ?  8  ?  6  ? 
7?   9? 

55.  Sixty-one  are  how  many  times  4?    5?    7?  6? 
8?    10?   9? 

56.  Seventy-nine  are  how  many  times  10  ?   9  ?  8  ? 
7?   6?  5? 

5* 


54  ARITHMETIC.  [PART  1. 

57.  Seventy  are  how  many  times  9?    5?    6?    8? 
7?    10? 

58.  Eighty-two  are  how  many  times  10?    7?    8? 
6?    9? 

59.  Sixty-six  are  how  many  times  9  ?   5?    6?    7? 
10?    8? 

60.  Eighty  are  how  many  times  10?    7?    6?    8? 
9? 

61.  Sixty-nine  are  how  many  times  9  ?  5?  7?  10? 
8?    6? 

62.  Eighty-one  are  how  many  times  10  ?    6  ?    8  ? 
7?   9? 

63.  Seventy-six  are  how  many  times  9  ?   5  ?    10  ? 
6?    7?    8? 

64.  Eighty-three  are  how  many  times  10  ?  6  ?  7 1 
9?    8? 

65.  Seventy-one  are  how  many  times  9  ?    5  ?    7  ? 
6?   8?    10? 

66.  Eighty-four  are  how  many  times  10  ?    6?    8? 
9?    7? 

67.  Seventy-seven  are  how  many  times  9?  7?  5? 
10?    8?    6? 

68.  Eighty-five  are  how  many  times  10?    8?    7? 
6?    9? 

69.  Ninety  are  how  many  times  9?    10?    6?    7? 
8? 

70.  Eighty-six  are  how  many  times  10?    9?    6? 
7?    8? 

71.  Ninety-four  are  how  many  times  9  ?    10  ?    8  ? 
6?    7? 

72.  Eighty-seven  are  how  many  times  10  ?  9  ?  7  ? 
6?    8? 

73.  Ninety-two  are  how  many  times  9  ?    10  ?    6  ? 
7?   8? 

74.  Eighty-eight  are  how  many  times  10  ?  9  ?  8  ? 
6?    7? 


SECT.  3.]  ARITHMETIC.  55 

75.  Ninety-five  are  Jiow  many  times  9?    10?    6? 
8?    7? 

76.  Eighty-nine  are  how  many  times  10?    9?   6? 
7?    8? 

77.  Ninety-eight  are  how  many  times  10  ?  9  ?  8  ? 
6?    7? 

78.  Ninety-four  are  how  many  times  9?    10?    6? 
8?    7? 

79.  One  hundred  are  how  many  times  6  ?  10  ?  8  ? 
9?    7? 

80.  Ninety-three  are  how  many  times  10  ?   6  ?  9  ? 
8?    7? 

81.  Ninety-nine  are  how  many  times  7?    10?   8? 
6?    9? 

82.  Ninety-six  are  how  many  times  9  ?  8  ?  7  ?  6  ? 
10? 

83.  Ninety-seven  are  how  many  times  10?  9?  6? 
'7?    8? 

D.   1.  If  an  orange  is  worth  3  apples,  how  many 
oranges  are  15  apples  worth? 

2.  In  8  pints,  how  many  quarts? 

3.  In  8  gills,  how  many  pints? 

4.  If  you   divide   twelve    apples   equally    among 
three  boys,  how  many  would  you  give  them  apiece  ? 

5.  How  many  hours  would  it  take  you  to  travel 
10  miles,  if  you  travel  three  miles  in  an  hour  ? 

6.  How  many  pence  are  there  in  eight  farthings  ? 

7.  How   many   pence    are   there   in    twelve    far- 
things ? 

8.  How  many  pence  are  there  in  seventeen  far- 
things ? 

9.  How  many  gallons  are  there  in  ten  quarts  ? 

10.  How  much  broadcloth,  at  6  dollars  a  yard, 
can  you  buy  for  seventeen  dollars? 

11.  How  many  pounds  of  raisins,   at  8  cents  a 
pound,  can  you  buy  for  twenty-five  cents? 


56  ARITHMETIC.  [PART  1. 

12.  In  twenty-eight  shillings,  how  many  dollars?* 

13.  In  twenty-eight  farthings,  how  many  pence? 

14.  How  many  barrels  of  flour,  at  7  dollars  a  bar- 
rel, can  you  buy  for  thirty-four  dollars? 

15.  How  many  reams  of  paper,  at  five  dollars  a 
ream,  can  you  buy  for  thirty-seven  dollars? 

1(3.  In  thirty-four  gills,  how  many  pints? 

17.  In  twenty-seven  quarts,  how  many  gallons? 

18.  If  an  orange  is  worth  six  apples,  how  many 
oranges  can  you  buy  for  forty  apples? 

19.  Thirty-six  shillings  are  how  many  dollars? 

20.  A  man  bought  thirty  apples  at  the  rate  of  3 
for  a  cent;  how  many  cents  did  they  come  to? 

21.  A   laborer   engaged   to   work  8   months   for 
ninety-six  dollars ;  how  much  did  he  receive  for  a 
month  ?     How  much  a  week,  allowing  4  weeks  to 
the  month  ?     How  many  shillings  a  day,  allowing  6 
working  days  to  the  week  ? 

22.  If  wine  is  worth  twenty  cents  a  pint,  what  is 
1  gill  worth  ? 

23.  If  you  can  buy  a  bushel  t  of  apples  for  forty 
cents,  what  is  the  price  of  a  peck  ? 

24.  If  you   buy  a  bushel  of  pears  for  forty-eight 
cents,  what  will  be  the  price  of  half  a  peck,  at  the 
same  rate? 

25.  Four  men  bought  a  horse  for  forty-eight  dol- 
lars ;  what  did  each  man  pay  ? 

26.  Five  men  bought  a  horse  for  seventy-five  dol- 
lars, and  sold   him  again  for  forty  dollars ;  what  did 
each  man  lose  by  the  bargain  ? 

27.  A  man  gave  sixty-three  cents  for   a  horse  to 
ride  nine  miles ;  what  was  the  price  of  one  mile's 
ride? 

28.  A  man  hired  a  horse  to  ride,  and  agreed  to 
give  8  cents  a  mile ;  he  paid  fifty-six  cents ;  how 
many  miles  did  he  ride  ? 

*  1  dollar  is  6  shillings.  1 1  bushel  is  4  nerk** 


SECT.  3.]  ARITHMETIC.  57 

29.  A  man  had  forty-two  dollars,  which  he  paid 
for  wood  at  7  dollars  a  cord ;  how  many  cords  did 
he  buy  ? 

30.  Two  boys  are  forty-eight  rods  apart,  and  fxrth 
running   the   same   way ;    but   the  hindermost  boy 
gains  upon  the  other  3  rods  in    a   minute  ;  in    how 
many  minutes  will  he  overtake  the  foremost  boy  ? 

31.  There    is    a    vessel    containing    sixty-three 
gallons  of  wine ;  it  has  a  pipe  which  discharges  7 
gallon^  in  an  hour ;  how  many  hours  will  it  take 
to  empty  the  vessel  ? 

32.  There   is   a  vessel    containing    eighty-seven 
gallons,  and  by  a  cock  ten  gallons  will  run  into  it  in 
an   hour;    in   how   many  hours   will  the   vessel  be 
filled? 

33.  If  one  man  can  do  a  piece  of  work  in  thirty 
days,  in  how  many  days  can  3  men  do  it  ?     In  how 
many  days  can  5  men  do  it  ? 

34.  If  you  wish  to  put  sixty-four  pounds  of  but- 
ter into  8  boxes,  how  many  pounds  would  you  put 
into  each  box? 

35.  If  you  had  seventy-two  pounds  of  butter,  which 
you  wished  to  put  into  boxes  containing  8  pounds 
each,  how  many  boxes  would  it  take  ? 

36.  If  a  man  can  perform  a  journey  in  thirty-six 
hours,  how  many  days  will  it  take  him  to  do  it  when 
the  days  are  nine  hours  long  ? 

37.  If  a  man  can  do  a  piece  of  work  in  forty-eight 
hours,  how  many  days  would  it  take  him  to  do  it,  if 
he  works  twelve  hours  in  a  day  ? 


58  ARITHMETIC.  [PART  1. 


SECTION    IV. 

A.   1.  AT  two  cents  a  yard,  what  will  3  yards  and 
one  half  of  a  yard  of  tape  cost  ? 

2.  3  times  2,  and  one  half  of  2  are  how  many? 

3.  At  three  dollars  a  yard,  what  will  4  yards  and  1 
third  of  a  yard  of  cloth  cost  ? 

4.  4  times  3,  and  1  third  of  three  are  how  many  ? 

5.  At  3  dollars  a  barrel,  what  will  3  barrels  and  2 
thirds  of  a  barrel  of  cider  cost  ? 

6.  3  times  3,  and  two  thirds  of  3  are  how  many  ? 

7.  If  a  man  earn  4  dollars  in  a  week,  how  many 
dollars  will  he  earn  in  3  weeks  and  1  fourth  of  a. 
week? 

8.  3  times  4,  and  1  fourth  of  4  are  how  many  ? 

9.  If  a  yard  of  cloth  cost  4  dollars,  what  will  5 
yards  and  3  fourths  of  a  yard  cost  ? 

10.  5  times  4,  and  3  fourths  of  4  are  how  many? 

11.  If  a  man  spend  five  dollars  in  a  week,  how 
many  dollars  will  he  spend  in  3  weeks  and  1  fifth  of 
a  week  ?     How  much  in  5  weeks  and  2  fifths  of  a 
week  ? 

12.  3  times  5,  and  1  fifth  of  5  are  how  many  ? 

13.  5  times  5,  and  2  fifths  of  5  are  how  many  ? 

14.  6  times  5,  and  3  fifths  of  5  are  how  many  ? 

15.  If  beer  is  worth  six  dollars  a  barrel,  what 
would  4  barrels  and  1  sixth  of  a  barrel  cost  ?     How 
much  would  7  barrels  and  5  sixths  of  a  barrel  cost  ? 

16.  4  times  6,  and  1  sixth  of  6  are  how  many  ? 

17.  7  times  6,  and  5  sixths  of  6  are  how  many  ? 

'  18.  At  7  dollars  a  barrel,  what  will  3  barrels  and 
1  seventh  of  a  barrel  of  flour  cost?  What  will  5 
barrels  and  2  sevenths  of  a  barrel  cost  ? 

19.  3  times  7,  and  1  seventh  of  7  are  how  many  ? 

20.  5  times  7,  and  2  sevenths  of  7  are  how  many  ? 


SECT.  4.]  ARITHMETIC.  59 

21.  8  times  5,  and  4  fifths  of  5  are  how  many  ? 

22.  8  times  6,  and  3  sixths  of  6  are  how  many? 

23.  At  8  dollars  a  yard,  what  will  4  yards  and  1 
eighth  of  a  yard  of  broadcloth  cost  ? 

24.  4  times  8,  and  1  eighth  of  8  are  how  many  ? 

25.  2  times  7,  and  3  sevenths  of  7  are  how  many  1 

26.  8  times  7,  and  4  sevenths  of  7  are  how  many  1 

27.  9  times  7,  and  6  sevenths  of  7  are  how  many  ? 

28.  3  times  8,  and  5  eighths  of  8  are  how  many  ? 

29.  9  times   8,    and   7   eighths   of   8    are   how 
many  ? 

30.  If  a  hundred  weight  of  sugar  cost  9  dollars, 
what  will  2  hundred  weight  and  1  ninth  of  a  hun- 
dred weight  cost  1    What  will  5  hundred  weight  and 
2  ninths  of  a  hundred  weight  cost  ? 

31.  2  times  9,  and  1  ninth  of  9  are  how  many  ? 

32.  5  times  9,  and  2  ninths  of  9  are  how  many  ? 

33.  6  times  9,  and  4  ninths  of  9  are  how  many  ? 

34.  2  times  10,  and  3  tenths  of  10  are  how  many? 

35.  7  times  9,  and  7  ninths  of  9  are  how  many  ? 

36.  5  times  10,  and  4  tenths  of  10  are  how  many  ? 

37.  8  times  9,  and  5  ninths  of  9  are  how  many  ? 

38.  4  times  10,  and  7  tenths  of  10  are  how  many? 

39.  6  times  10,  and  9  tenths  of  10  are  how  many? 

15.  1.  A  man  bought  2  oranges  at  6  cents  apiece  ; 
how  many  cents  did  they  come  to?  He  paid  for 
them  with  cherries  at  4  cents  a  pint;  how  many 
pints  did  it  take  ? 

2.  2  times  6  are  how  many  times  4? 

3.  A  man  bought  3  yards  of  cloth  at  4  dollars 
a  yard;  how  many  dollars  did   it  come  to?    How 
much   flour    at  6  dollars  a  barrel  would  it  take  to 
pay  for  it  ? 

4.  3  times  4  are  how  many  times  6? 

5.  A  man  bought  4  peaches  at  3  cents  apiece ; 
how  many  cents  did  they  come   to  ?    He  paid  for 


60  ARITHMETIC.  [PART  1. 

them  with  pears  at  2  cents  apiece ;  how  many  pears 
did  it  take  1 

6.  4  times  3  are  how  many  times  2  ? 

7.  Bought  2  hundred  weight  of  sugar  at  9  dol- 
lars a  hundred  weight,  and  paid  for  it  with  wood 
at  6  dollars  a  cord ;  how  many  cords  did  it  take  ? 

8.  2  times  9  are  how  many  times  6  ? 

9.  Bought  3  barrels  of  flour  at  8  dollars  a  barrel, 
and  paid  for  it  with  cider  at  4  dollars  a  barrel ;    how 
many  barrels  did  it  take  ? 

10.  3  times  8  are  how  many  times  4? 

11.  12  times  3  are  how  many  times  5? 

12.  6  times  4  are  how  many  times  8? 

13.  3  times  10  are  how  many  times  6  ? 

14.  4  times  9  are  how  many  times  6  ? 

15.  How  much  flannel,  worth  4  shillings  a  yard,, « 
must  be  given  for  3  yards  of  silk,  worth  5  shillings  a 
yard? 

16.  3  times  5  are  how  many  times  4  ? 

17.  2  times  7  are  how  many  times  3?  5?  4? 

18.  4  times  5  are  how  many  times  3  ?  6?  7? 

19.  3  times  7  are  how  many  times  4?5?6?8? 
9? 

20.  Bought  2  kegs  and  2  sevenths  of  a  keg  of 
tobacco  at  7  dollars  a  keg,  and   paid   for  it  with 
wood  at  4  dollars  a  cord ;  how  many  cords  did  it 
take  ?    How  much  butter,  at  3  dollars  a  box,  would 
it  take  to  pay  for  it  ? 

21.  2  times  7,  and  2  sevenths  of  7  are  how  many 
times  4?  3?  5?  6?  8? 

22.  Bought  3  bushels  and  3  fifths  of  a  bushel  of 
corn  at  5  shillings  a  bushel,  and  paid  for  it  with 
wheat  at  6  shillings  a  bushel ;  how  many  bushels  of 
wheat  did  it  take  ? 

23.  3  times  5,  and  3  fifths  of  five  are  how  many 
timesG?  9?  4?  7?  3?  8? 

24.  How  much  sugar,  that  is  8  dollars  a  hundred 


SECT.  4.]  ARITHMETIC.  61 

weight,  can  be  bought  for  4  cords  and  2  sevenths  of 
a  cord  of  wood,  at  7  dollars  a  cord  ? 

25.  4  times  7,  and  2  sevenths  of  7  are  how  many 
times  6?  8?  5?  3?  9?  10? 

26.  5  times  5,  and  3  fifths  of  5  are  how  many 
times  4?  8?  9?  7?  10?  3?  6? 

27.  6  times  7,  and  3  sevenths  of  7  are  how  many 
times  9?  4?  5?  8?  10? 

28.  5  times  8,  and  3  eighths  of  8  are  how  many 
times  6?  9?  4?  7?  10? 

29.  7  times  8,  and  5  eighths  of  8  are  how  many 
times  9?  6?  10?  4?  5? 

30.  5  times  9,  and  4  ninths  of  9  are  how  many 
times  7?  8?  6?  10?  4? 

31.  7  times  9,  and  7  ninths  of  9  are  how  many 
times  6?  8?  10?  5?  4? 

32.  6  times  10,  and  3  tenths  of  10  are  how  many 
times  7?  5?  4?  9?  8? 

33.  8  times  10,  and  4  tenths  of  10  are  how  many 
times  6?  7?  9? 

34.  8  times  9,  and  3  ninths  of  9  are  how  many 
times  6?  10?  7? 

.  C.   1.  Bought  4  bushels  of  apples,  at  3  shillings  a 
bushel ;  how  many  dollars  did  they  come  to  ? 

2.  How  many  apples,  at  2  cents  apiece,  must  you 
give  for  2  lemons  at  4  cents  apiece  ? 

3.  How  many  pears,  at  3  cents  apiece,  must  you 
give  for  3  oranges  at  5  cents  apiece  ? 

4.  How  many  barrels  of  cider,  at  3  dollars  a  bar- 
rel, must  be  given  for  5  boxes  of  butter,  at  4  dollars 
a  box? 

5.  A  man  bought  4  yards  of  broadcloth,  at  7  dol- 
lars a  yard,  and  paid  for  it  with  flour,  at  5  dollars  a 
barrel ;  how  many  barrels  did  he  give  ? 

6.  If  2  apples  cost  4  cents,  what  would  4  apples 
cost? 

6 


62  ARITHMETIC.  [PART  1. 

7.  If  3  apples  are  worth  6  cents,  how  many  ap- 
ples must    you  give  for  8  pears,  that  are   worth  3 
cents  apiece  ? 

8.  James  had  8  oranges  that  were  worth  5  cents 
apiece,  and   George  had  5  quarts  of  cherries  that 
were  worth  6  cents  a  quart,  which  he  gave  to  James 
for  a  part  of  his  oranges ;  how  many  oranges  did  he 
buy,  and  how  many  had  James  left? 

9.  Bought  8  yards  of  cloth,  at  9  shillings  a  yard ; 
how  many  dollars  did  it  come  to? 

10.  Bought  5  bushels  and  3  sevenths  of  a  bushel 
of  salt,  at  7  shillings  a  bushel ;  how  many' dollars  did 
it  come  to  ? 

11.  Bought  9  boxes  and  2  thirds  of  a  box  of  rai- 
sins for  3  dollars  a  box,  and  paid  for  it  with  cider 
at  4  dollars  a  barrel ;  how  many  barrels  did  it  take? 

12.  Bought  8  pounds  and  4  sevenths  of  a  pound 
of  opium  at  7  dollars  a  pound,  and  paid  for  it  witli 
cloth  at  5  dollars  a  yard;  how  many  yards  did  it 
take? 

13.  Bought  6  chaldrons  and  4  ninths  of  a  chaldron 
of  coal  at  9  dollars  a  chaldron,  and  paid  for  it  with 
oranges  at  5  dollars  a  box ;  how  many  boxes  did  it 
take? 

14.  Bought   7  cases  and   5  sixths  of   a  case  of 
Florence   oil  at  6  dollars  a  case,    and    paid  for    it 
with   sheet   lead    at   7   dollars    a   hundred  weight  ; 
how  many  hundred  weight  did  it  take  ? 

15.  How  many  dozen  of  eggs,  at  8  cents  a  dozen, 
must  you  give  for  7  pounds  of  sugar,  at  10  cents  a 
pound  ? 

16.  How  much   barley,  at  3  shillings  a  bushel, 
must  be  given  for  8  bushels  of  wheat,  at  7  shillings 
a  bushel  1 

17.  How  much  cloth,  at  4  shillings  a  yard,  must 
be  given  for  a  firkin  of  butter,  worth  8  dollars  ? 

18.  How  much  cloth,  at  5  shillings  a  yard,  can  be 


SECT.  5.]  ARITHMETIC.  63 

bought  for  2  reams  of  paper,  at  5  dollars  a  ream'? 

19.  How  much  wheat,  at  7  shillings  a  bushel,  can 
be  bought  for  2  barrels  of  cider,  at  4  dollars  and  a 
half  a  barrel  ? 

20.  How  long  would  it  take  a  man  to  lay  up  10 
dollars,  if  he  saves  4  shillings  a  day? 

21.  If  a  man  earn  8  shillings  a  day,  how  many 
dollars  would  he  earn  in  10  days  ? 

22.  A  man  bought  twenty  pears  at  the  rate  of  2 
for  3  cents  ;  how  much  did  they  come  to  ? 

23.  How  many  eggs,  at  the  rate  of  3  for  5  cents, 
can  you  buy  for  thirty  cents  1 

24.  A  man  hired  a  laborer,  and  agreed  to  give  him 
5  dollars  for  every  3  days'  work  ;  how  much  did  he 
give  him  a  week,  there  being  6  working  days  in  a 
week  ?  How  much  was  it  a  month,  allowing  4  weeks 
to  the  month  ? 

25.  If  a  man  receives  5  dollars  for  3  days'  work, 
how  many  shillings  is  that  a  day  ? 

26.  5  men  bought  a  horse  for  sixty-three  dollars, 
and  paid  two  dollars  a  week  for  keeping  him;    at 
the  end  of  8   weeks,  they  sold   him   for   fifty-four 
dollars ;  how  much  did  each  man  lose  by  the  bar- 
gain ? 


SECTION  V. 

A.  1.  JAMES  had  4  apples,  and  John  had  half  as 
many  ;  how  many  had  he  ? 

2.  If  an  orange  cost  6  cents,  and  an  apple  half  as 
much,  how  much  does  the  apple  cost? 

3.  If  you   divide   8   apples  equally  between  two 
boys,  what  part  of  them  must  each  have  ? 

Ans.  One  half  of  them. 


C4  ARITHMETIC.  [PART  1. 

4.  What  is  1  half  of  8? 

5.  If  you  divide  8  apples  equally  among  4  boys, 
what  part  of  them  must  each  have? 

Ans.  One  fourth  of  them. 

6.  What  is  1  fourth  of  8? 

7.  If  you  divide  6  oranges  equally  among  3  boys, 
what  part  of  them  must  1  boy  have  ? 

8.  What  is  1  third  of  6? 

9.  If  4  yards  of  cloth  cost  8  dollars,  what  part  of 
8  dollars  would  1  yard  cost?  What  part  of  8  dollars 
would  2  yards  cost  ?  What  part  of  8  dollars  would  3 
yards  cost? 

10.  What  is  1  fourth  of  8  ?    What  is  2  fourths  of 
8?    What  is  3  fourths  of  8? 

11.  If  6  yards  of  cloth  will  make  3  coats,  what 
part  of  6  yards  will  make  1  coat  ?   What  part  of  6 
yards  will  make  2  coats  ? 

12.  What  is  one  third  of  6  ?    What  is  two  thirds 
of6? 

13.  If  3  barrels  of  cider  cost  9  dollars,  what  part 
of  9  dollars  will  1  barrel  cost  ?   What  part  of  9  dol- 
lars will  2  barrels  cost  ? 

14.  What  is  1  third  of  9?    What  is  2  thirds  of  9? 

15.  If  2  yards  of  cloth  cost  10  dollars,  what  part 
of  10  dollars  will  1  yard  cost?    What  part  of  10  dol- 
lars will  3  yards  cost  ? 

16.  What  is  1  half  of  10?   What  is  3  halves  of 
10? 

17.  If  2  barrels  of  flour  cost  twelve  dollars,  what 
part  of  twelve  dollars  will  one  barrel  cost?    What 
part  of  twelve   dollars  will  3  barrels  cost  ?    What 
part  of  twelve  dollars  will  5  barrels  cost  ? 

18.  What  is  1  half  of  twelve  ?   What  is  3  halves 
of  12  ?   What  is  5  halves  of  twelve  ? 

19.  If  4  barrels  of  cider  cost  twelve  dollars,  what 
part  of  twelve   dollars  will    1  barrel   cost?    What 
part  of  twelve  dollars  will  3  barrels  cost?    What 


SECT.  5.]  ARITHMETIC.  65 

part  of  twelve  dollars  will  5  barrels  cost?    What 
part  of  twelve  dollars  will  7  barrels  cost? 

20.  What  is  1  fourth  of  twelve  ?  What  is  2  fourths 
of  twelve?  3  fourths?  5  fourths?  7  fourths? 

21.  If  3  oranges  cost  twelve  cents,  what  part  of 
twelve  cents  wrill  1  orange  cost  ?  What  part  of  twelve 
cents  will   2  oranges  cost?    What   part    of  twelve 
cents   will  4  oranges   cost?    What  part   of  twelve 
cents  will  5  oranges  cost?  7  oranges?  10  oranges? 

22.  What  is  1  third  of  twelve  ?  2  thirds  1  4  thirds  ? 

5  thirds?  7  thirds?  10  thirds? 

23.  If  5  bushels  of  wheat  cost  10  dollars,  what 
part,  of  10  dollars  will  1  bushel  cost?    What  part  of 
10  dollars  will  2  bushels  cost  ?  3  bushels  ?  4  bushels  ? 

6  bushels  ?  7  bushels  ? 

24.  What  is  1  fifth  of  10?   2  fifths?   3  fifths?  4 
fifths  ?  6  fifths  ?  7  fifths  ? 

25.  What  is  1  half  of  fourteen?    1  seventh?    2 
sevenths  ?  3  sevenths  ?  5  sevenths  ? 

26.  What  is  1  third  of  fifteen?   2  thirds?  1  fifth? 

2  fifths  ?  3  fifths  ?  4  fifths  ? 

27.  What  is  1  half  of  sixteen?  3  fourths?  1  eighth  ? 

3  eighths  ?  5  eighths  ?  7  eighths  ?  1  sixteenth  ? 

28.  What  is  1  half  of  eighteen  ?  2  thirds?  1  sixth? 
5  sixths  ?    1  ninth  ?   2  ninths  ?   4  ninths  ?   5  ninths  ? 

7  ninths  ?  8  ninths  ?   1  eighteenth  ?  5  eighteenths  ? 

29.  What  is  1  half  of  twenty  ?  1  fourth  ?  3  fourths  ? 
1  fifth?    3  fifths?    2  fifths?    4  fifths?     1  twentieth? 
3  twentieths  ?  7  twentieths  ? 

30.  What  is  1  third  of  twenty-one  ?   2  thirds  ?  1 
seventh  ?    5  sevenths  ?    3  sevenths  ?    6  sevenths  ?   4 
sevenths  ? 

31.  What  is  1  half  of  twenty-two?    1  eleventh? 
3  elevenths?    5  elevenths?    7  elevenths?    10  elev- 
enths ? 

32.  What  is  1  half  of  twenty-four  ?    1  third?   2 
thirds?  1  fourth?   3  fourths  ?    1  sixth  ?  5  sixths?    1 

6* 


66  ARITHMETIC.  [PART  1. 

eighth?  3 eighths?  7 eighths?  1  twelfth?  5 twelfths? 
7  twelfths  ? 

33.  What  is  1  half  of  twenty-six  ? 

34.  What  are  2  thirds  of  twenty-seven  ?  2  ninths  ? 
4  ninths  ?  5  ninths  ?  8  ninths  ?  7  ninths  ? 

35.  What  is  1  half  of  twenty-eight  ?  3  fourths  ?  2 
sevenths?  5  sevenths?  3  sevenths? 

36.  What  is  1  sixth  of  thirty?   3  fifths?  7 tenths? 
2  thirds?  1  half? 

37.  What  is  1  half  of  thirty-two?    3  fourths,?   5 
eighths?  1  sixteenth? 

38.  What  is  1  half  of  thirty-four  ? 

39.  What  is  5  sixths  of  thirty-six?    2  thirds?    1 
half?  7  twelfths?   4  ninths?   3  fourths? 

40.  What  is  1  half  of  thirty-eight  ? 

41.  What  is  1  third  of  thirty-nine  ? 

42.  What  is    3  fourths  of  forty  ?    7  eighths  ?    9 
tenths?  1  half? 

43.  What  is  3  sevenths  of  forty-two?  5  sixths? 

B.  1.  A  boy,  having  twelve  apples,  kept  1  fourth 
of  them  himself,  and  divided  the  other  3  fourths  of 
them  equally  among  4  of  his  companions  ;  how  many 
did  he  give  them  apiece  ? 

2.  3  fourths  of  twelve  are  how  many  times  4  ? 

3.  A  man,  having  fourteen  bushels  of  grain,  di- 
vided 5  sevenths  of  it  equally  among  3  men ;  how 
much  did  he  give  them  apiece  ? 

4.  5  sevenths  of  fourteen  are  how  many  times  3? 

5.  A  man,  having  fifteen  shillings,  gave  away  4 
fifths  of  it ;  how  many  dollars  did  he  give  away  ? 

6.  4  fifths  of  fifteen  are  how  many  times  6  ? 

7.  A  man,  having  twenty-one  cents,  paid  away  6 
sevenths  of  them  for  oranges  at  5  cents  apiece ;  how 
many  oranges  did  he  buy  ? 

8.  6  sevenths  of  twenty-one  are  how  many  times 
5? 


SECT.  5.]  ARITHMETIC.  67 

9.  A  man  bought  a  piece  of  cloth  for  twenty-four 
shillings,  and  sold  it  again  for  5  thirds  of  what  he 
gave  for  it;  how  many  dollars  did  he  sell  it  for? 

10.  5  thirds  of  twenty-four  are  how  many  times 
6? 

11.  7  fifths  of  twenty  are  how  many  times  6  ? 

12.  5  sevenths  of  thirty-five  are  how  many  times 
8? 

13.  7  sixths  of  thirty-six  are  how  many  times  5? 

14.  9  sevenths  of  forty  two  are  how  many  times 
8? 

15.  7  ninths  of  forty-five  are  how  many  times  6  ? 

16.  5  sixths  of  forty-eight  are  how  many  times  7 1 

17.  6  fifths  of  fifty  are  how  many  times  9? 

18.  8  ninths  of  fifty- four  are  how  many  times  5  ? 

19.  9  sevenths   of  fifty-six  are   how   many  times 
10? 

20.  7  sixths  of  sixty  are  how  many  times  8  ? 

21.  9  sevenths  of  sixty-three  are  how  many  times 
7? 

22.  10  eighths  of  sixty-four  are  how  many  times 
9? 

23.  6  eighths  of  seventy-two  are  how  many  times 
7? 

24.  4  sevenths  of  eighty-four  are  how  many  times 
9? 

25.  7  tenths  of  ninety  are  how  many  times  8  ? 

C.   1.  Charles  had  6  apples,  and  gave  1  third  of 
them  to  John  ;  how  many  did  he  give  him  ? 

2.  Albert  had  9  cents,  and  spent  2  thirds  of  them ; 
how  many  had  he  left? 

3.  James  had  10  pears,  and  gave  1  half  of  them 
to  one  of  his  companions,  and  2  fifths  of  them  to 
another  ;  how  many  did  he  give  away  ? 

4.  If  3  yards  of  cloth  cost  6  dollars,  what  is  that 
a  yard  1 


68  ARITHMETIC.  [PART  1. 

5.  If  4  yards  of  cloth  cost  twelve  dollars,  what 
will  2  yards  cost  ? 

6.  If  9  apples  cost  eighteen  cents,  what  will  3 
apples  cost  ? 

7.  If  3  oranges  cost  18  cents,  what  will  2  cost  ? 

8.  James  had  twenty-five  cents,  and  he  gave  4 
fifths  of  them  for  10  apples ;  how  much  did  he  give 
for  all  the  apples  ?  how  much  apiece  1 

9.  A  man  had  thirty  dollars,  and  gave  5  sixths  of 
them  for  8  yards  of  cloth ;  how  much  did  he  give 
a  yard  ? 

10.  A  man   had  forty  yards  of  cloth,  and  sold  3 
fifths  of  it  for  twenty-four  dollars ;  what  was  that  a 
yard? 

11.  A  man   had   forty-two   barrels  of  flour,  and 
sold  2  sevenths  of  it  for  6  dollars  a  barrel ;    how 
much  did  it  come  to  ? 

12.  A  boy  had   sixty-three  nuts,   and  divided  4 
sevenths  of  them  equally  among  six  of  his  compan- 
ions ;  how  many  did  he  give  them  apiece  ? 

13.  If  4  yards  of  cloth  cost  twelve  dollars,  what 
will  3  yards  cost  ? 

14.  If  5  oranges  cost  twenty-five  cents,  what  will 
3  cost  ? 

15.  If  3  oranges  cost  fifteen  cents,  what  will  7  cost  ? 

16.  If  3  barrels  of  cider  cost  twelve  dollars,  what 
will  10  barrels  cost? 

17.  If  7   pounds  of  flour  cost   thirty-five   cents, 
what  will  9  pounds  cost  ? 

18.  If  5  firkins  of  butter  cost  forty  dollars,  what 
will  3  firkins  cost  ? 

19.  If  2  men   can  do  a  piece  of  work  in  6  days, 
how  long  would  it   take   4   men   to  do   the  same 
work  ? 

20.  If  6  men  can  do  a  piece  of  work  in  twelve 
days,  in  how  many  days  will  3  men  do  the  same 
work? 


SECT.  5.]  ARITHMETIC.  69 

21.  If  3  men  can  do  a  piece  of  work  in  twelve  days, 
in  how  many  days  will  4  men  do  the  same  work  ? 

22.  If  2  cocks  of  a  certain  size  will  empty  a  cis- 
tern in  6  hours,  in  how  long  a  time  will  3  cocks  of 
the  same  size  empty  it  ? 

23.  Three  men,  setting  out  on  a  journey,  pur- 
chased 5  loaves  of  bread  apiece,  but  before  they  had 
eaten  any  of  it,  two  other  men  joined  them,  and 
they  agreed  to  share  the  bread  equally  among  the 
whole ;  how  many  loaves  did  they  have  apiece  ? 

24.  If  4  barrels  of  flour  cost  twenty-four  dollars, 
what  would  7  barrels  cost  ?    How  much  cider,  at  3 
dollars  a  barrel,  would  7  barrels  of  flour  buy  1 

25.  A  man  bought  a  quantity  of  flour  for  fifty- 
four  dollars  ;  and  another  man  gave  him  9  yards  of 
cloth  for  5  sixths  of  it ;  what  was  the  cloth  worth 
a  yard  1 

26.  If  9  yards  of  cloth  cost  fifty-four  dollars,  how 
many  boxes  of  butter,  at  4  dollars  a  box,  would  5 
yards  of  the  same  cloth  buy  1 

27.  Bought   8  firkins  of  butter   for   seventy-two 
dollars,  and  gave  6  of  them  for  7  yards  of  cloth ; 
what  was  a  yard  of  the  cloth  worth  ? 

28.  A  man  bought   6  barrels  of  flour  for   sixty 
dollars,  and  gave  4  barrels  of  it  for  cider  at  5  dol- 
lars a  barrel ;  how  many  barrels  did  he  buy  ? 

Note.  The  manner  of  writing  numbers  with  fig- 
ures has  been  explained  as  far  as  ten.  The  numbers 
from  ten  to  one  hundred  are  written  as  follows: — • 

Ten  is  written 10 

Eleven 11 

Twelve    . 12 

Thirteen 13 

Fourteen 14 

Fifteen 15 

Sixteen    :  .  16 

Seventeen  .  ...  17 


70  ARITHMETIC.  [PART  1 

Eighteen 18 

Nineteen 19 

Twenty 20 

Twenty-one 21 

Twenty-two 22 

Twenty-three 23 

Twenty-four 24 

Twenty-five 25 

Twenty-six 26 

Twenty-seven 27 

Twenty-eight 28 

Twenty-nine 29 

Thirty 30 

Thirty-one,  &c 31 

Forty 40 

Fifty 50 

Sixty 60 

Seventy 70 

Eighty 80 

Ninety 90 

One  hundred 100 


SECTION  VI. 

A.  1.  BOUGHT  1  half  of  a  yard  of  cloth  for  1 
shilling ;  what  would  be  the  price  of  a  yard  at  the 
same  rate  ? 

2.  If  1  half  pint  of  cherries  cost  2  cents,  what 
will  a  pint  cost  ? 

3.  If  1  fourth  of  a  barrel  of  flour  cost  2  dollars, 
what  would  a  barrel  cost  ? 

4.  2  is  1  half  of  what  number  ?    1  fourth  of  what 
number  ? 


SECT.  6.]  ARITHMETIC.  71 

5.  If  1  third  of  a  yard  of  cloth  cost  2  dollars,  what 
is  that  a  yard  ? 

6.  2  is  1  third  of  what  number  ? 

7.  If  1  third  of  a  yard  of  cloth  cost  3  dollars,  what 
is  that  a  yard  ? 

<8.  3  is  1  third  of  what  number  1 

9.  If  1  fourth  of  a  firkin  of  butter  cost  3  dollars, 
what  is  that  a  firkin  ? 

10.  3  is  1  fourth  of  what  number  1 

11.  A  man  bought  1  third  of  a  bushel  of  wheat 
for  4  shillings ;  what  would   a  bushel  cost  at  that 
rate? 

12.  4  is  1  third  of  what  number  ? 

13.  If  a  man  can  ride  2  miles  in  I  fifth  of  an  hour, 
how  far  can  he  ride  in  an  hour  ? 

14.  2  is  1  fifth  of  what  number  ? 

15.  A  man,  being  asked  the  age  of  his  eldest  son, 
answered  that  his  youngest  son,  who  was  3  years  old, 
was  just  1  fifth  of  the  age  of  his  eldest  son ;  how  old 
was  the  eldest  son  ? 

16.  3  is  one  fifth  of  what  number? 

17.  A  man    bought    1    sixth  part  of  a  hundred 
weight  of  sugar  for  2  dollars ;  what  would  a  hundred 
weight  cost  at  the  same  rate  ? 

18.  2  is  1  sixth  of  what  number  ? 

19.  Bought  1  fifth  of  a  pound  of  starch  for  5  cents, 
what  was  that  a  pound  ? 

20.  5  is  1  fifth  of  what  number  ? 

21.  Bought  1  fourth  of  a  pound  of  aqua  fortis  for 
6  cents;  what  was  that  a  pound  ? 

22.  6  is  1  fourth  of  what  number  ? 

23.  Bought  1  seventh  of  a  hundred  weight  of  co- 
coa for  4  dollars ;  what  would  1  hundred  weight  cost 
at  the  same  rate  ? 

24.  4  is  1  seventh  of  what  number  ? 

25.  7  is  1  fifth  of  what  number  ? 

26.  5  is  1  third  of  what  number  ? 


72  ARITHMETIC.  [?JLRT  1. 

27.  4  is  1  eighth  of  what  number  ? 

28.  6  is  1  sixth  of  what  number  ? 

29.  8  is  1  third  of  what  number  ? 

30.  9  is  1  fourth  of  what  number  ? 

31.  7  is  1  sixth  of  what  number? 

32.  8  is  1  seventh  of  what  number  ? 

33.  9  is  1  eighth  of  what  number  1 

34.  8  is  1  tenth  of  what  number  ? 

35.  7  is  1  ninth  of  what  number? 

36.  6  is  1  fifth  of  what  number  ? 

37.  10  is  1  seventh  of  what  number  1 

B.  1.  A  man  bought  some  linen  and  some  cotton 
cloth  ;  for  the  linen  he  gave  4  shillings  a  yard,  which 
was  twice  as  much  as  he  gave  for  the  cotton ;  what 
did  he  give  for  a  yard  of  the  cotton  ? 

2.  4  is  2  times  what  number  ? 

3.  If  2  thirds  of  a  yard  of  cloth  cost  6  dollars, 
what  would  1  third  cost? 

4.  6  is  2  times  what  number  ? 

5.  If  3  fourths  of  a  barrel  of  flour  cost  6  dollars, 
what  will  one  fourth  of  a  barrel  cost  ? 

6.  6  is  three  times  what  number  ? 

7.  If  2  fifths  of  a  pound  of  chocolate  cost  8  cents, 
what  would  1  fifth  of  a  pound  cost  ? 

8.  8  is  2  times  what  number  ? 

9.  If  3  fifths  of  a  pound  of  candles  cost  9  cents, 
what  will  1  fifth  of  a  pound  cost  ? 

10.  9  is  3  times  what  number  ? 

11.  If  2  sevenths  of  a  pound  of  spermaceti  can- 
dles cost  10  cents,  what  will  1  seventh  of  a  pound 
cost? 

12.  10  is  2  times  what  number  ? 

13.  If  5  eighths  of  a  pound  of  cotton   cost  10 
cents,  what  will  1  eighth  cost  ? 

14.  10  is  five  times  what  number  ? 

15.  If  2  thirds  of  a  yard  of  cloth  cost  4  dollars, 


SECT.  6.]  ARITHMETIC.  73 

what  will  one  third  cost  ?    If  one  third  of  a  yard  cost 
2  dollars,  what  will  a  yard  cost  ?* 

16.  If  4  is  2  thirds  of  some  number,  what  is  1 
third  of  the  same  number? — 2  is  I   third  of  what 
number  ?* — Then  4  is  two  thirds  of  what  ? 

17.  If  2  thirds  of  a  barrel  of  flour  cost  6  dollars, 
what  will  1  third  of  a  barrel  cost  ?    If  1  third  of  a 
barrel  cost  3  dollars^  what  will  a  barrel  cost?" 

18.  If  6  is  2  thirds  of  some  number,  what  is  1 
third  of  the  same  number? — 3  is  one  third  of  what 
number  ? — Then  6  is  two  thirds  of  what  ? 

19.  If  3  fourths  of  a  bushel  of  wheat  cost  6  shil- 
lings, what  will  1   fourth  of  a   bushel  cost?     If  1 
fourth   of  a   bushel    cost   2  shillings,   what  will    a 
bushel  cost  ? 

20.  If  6  is  3  fourths  of  some  number,  what  is  1 
fourth  of  the  same  number  ? — 2  is  1  fourth  of  what 
number? — Then  6  is  3  fourths  of  what? 

21.  If  2  fifths  of  a  gallon  of  wine  cost  4  shillings, 
what  will  1  fifth  of  a  gallon  cost  ?    If  one  fifth  of  a 
gallon  cost  2  shillings,  what  will  a  gallon  cost  ? 

22.  If  4  is  two  fifths  of  some  number,  what  is  1 
fifth  of  the  same  number  ? — 2   is    1   fifth  of  what 
number  ?— Then  4  is  2  fifths  of  what  ? 

23.  If  3  sevenths  of  a  pound  of  tobacco  cost  6 
cents,  what  will  1  seventh  of  a  pound  cost  ?     If  1 
seventh  of  a  pound  cost  2  cents,  what  will  a  pound 
cost? 

24.  If  6  is  3  sevenths  of  some  number,  what  is  1 
seventh  of  the  same   number  ? — 2  is  1  seventh  of 
what  number  ? — Then  6  is  3  sevenths  of  what  ? 

25.  If  2  sevenths  of  a  barrel  of  fish  cost  4  dol- 
lars, what  will  1   seventh  of  a  barrel  cost?    What 
will  a  barrel  cost  ? 

26.  4  is  2  sevenths  of  what  number  ? 

27.  If  3  eighths  of  a  pound  of  chocolate  cost  6 

*See  this  Sect  Art.  A, 


74  ARITHMETIC.  [PART  1. 

cents,  what  will  1  eighth  of  a  pound  cost  ?    What 
will  a  pound  cost? 

28.  6  is  3  eighths  of  what  number  ? 

29.  If  eight  cents  will  buy  2  fifths  of  a  pound  of 
aqua  fortis,  how  many  cents  will  buy  a  pound  ? 

30.  8  is  2  fifths  of  what  number  ? 

31.  A  man  bought  3  fourths  of  a  hundred  weight 
of  yellow  ochre  for  9  dollars  ;  what  was  that  a  hun- 
dred weight  ? 

32.  9  is  three  fourths  of  what  number? 

33.  8  is  4  ninths  of  what  number  ? 

34.  9  is  3  tenths  of  what  number  ? 

35.  10  is  5  sevenths  of  what  number  ? 

36.  12  is  3  fifths  of  what  number? 

37.  12  is  4  ninths  of  what  number? 

38.  10  is  2  sevenths  of  what  number  ? 

39.  14  is  7  fifths  of  what  number? 

40.  15  is  3  elevenths  of  what  number? 

41.  16  is  2  fifths  of  what  number? 

42.  18  is  6  tenths  of  what  number  ? 

43.  20  is  5  ninths  of  what  number  ? 

44.  21  is  3  ninths  of  what  number? 

45.  24  is  8  ninths  of  what  number? 

C»  1 .  If  5  eighths  of  a  cask  of  claret  wine  cost  15 
dollars,  what  is  that  a  cask  ?  How  much  cider,  at  4 
dollars  a  barrel,  would  it  take  to  pay  for  a  cask  of 
the  wine  ? 

2.  15  is  five  eighths  of  how  many  times  4  ? 

3.  If  2  thirds  of  a  pound  of  coffee  cost  18  cents, 
how  much  would  a  pound  cost?  How  many  oranges, 
at  5  cents  apiece,  might  be  bought  for  a  pound  ? 

4.  18  is  2  thirds  of  how  many  times  5  ? 

5.  A  man  bought  4  sevenths  of  a  hundred  weight 
of  sugar  for  20  shillings;  how  many  dollars  would  a 
hundred  weight  come  to  at  the  same  rate  ? 

6.  20  is  4  seventh?  of  how  many  times  6? 


SECT.  6.]  ARITHMETIC.  75 

7.  A  man  sold  a  cow  for  21  dollars,  which  was 
only  seven  tenths  of  what  she  cost  him  ;  how  much 
did  she  cost  him  ?    When  he  bought  her,  he  paid 
for  her  with  cloth  at  8  dollars  a  yard ;  how  many 
yards  of  cloth  did  he  give  ? 

8.  21  is  7  tenths  of  how  many  times  8  ? 

9.  A  man,  being  asked  the  age  of  his  youngest 
son,  answered,  that  the  age  of  his  eldest  son  was  24 
years,  which  was  3  fifths  of  his  own  age ;  and  that  his 
own  age  was  10  times  as  much  as  that  of  his  young- 
est son ;  what  was  his  age  ?  and  what  was  the  age 
of  his  youngest  son  ? 

10.  24  is  3  fifths  of  how  many  times  10? 

11.  27  is  3  fifths  of  how  many  times  7  ? 

12.  28  is  7  tenths  of  how  many  times  9  ? 

13.  30  is  5  eighths  of  how  many  times  7  ? 

14.  32  is  4  sevenths  of  how  many  times  61 

15.  36  is  9  eighths  of  how  many  times  5? 

16.  40  is  8  ninths  of  how  many  times  8  ? 

17.  42  is  6  fifths  of  how  many  times  4  ? 

18.  45  is  9  eighths  of  how  many  times  6 1 

19.  48  is  8  ninths  of  how  many  times  7 1 

20.  50  is  5  sevenths  of  how  many  times  8  ? 

21.  54  is  9  sixths  of  how  many  times  7  ? 

22.  56  is  7  ninths  of  how  many  times  10? 

23.  60  is  10  sevenths  of  how  many  times  4  ? 

24.  63  is  9  eighths  of  how  many  times  5  ? 

25.  64  is  8  ninths  of  how  many  times  7  ? 

26.  70  is  10  sevenths  of  how  many  times  8  ? 

27.  72  is  9  fifths  of  how  many  times  6  ? 

28.  SO  is  10  thirds  of  how  many  times  4? 

29.  80  is  8  fifths  of  how  many  times  6  ? 


I>.   1.  A  boy  gave  away  4  cents,  which  was  1 
third  of  all  he  had  ;  how  many  had  he  at  first  ? 
2,  A  boy  gave  5  apples  to  one  of  his  companions, 


76  ARITHMETIC.  [PART  1. 

which  was    I  fourth  of  what  he   had ;    how  many 
had  he  ? 

3.  A  man  paid  away  4  dollars,  which  was  2  thirds 
of  all  the  money  he  had;  how  much  had  he  ? 

4.  A  man  sold  a  watch  for  18  dollars,  which  was 
3  fourths  of  what  it  cost  him  ;  how  much  did  it  cost '? 

5.  A  man  sold  a  cow  for  15  dollars,  which  was  3 
fifths  of  what  the  cow  cost ;  how  much  did  he  lose 
by  his  bargain  ? 

6.  A  man  bought  12  yards  of  cloth,  and  sold  it 
for  54  dollars,  which  was  0  eighths  of  what  it  cost 
him ;  what  did  it  cost  him  a  yard  ?  and  how  much 
did  he  gain  by  his  bargain  1 

7.  There  is  a  pole  standing  in  the  water,  so  that 
10  feet  of  it  is  above  the  water,  which  is  2  thirds  of 
the  whole  length  of  the  pole  ;  how  long  is  the  pole  ? 

8.  There  is  a  pole  two  thirds  under  water,  and  4 
feet  out ;  how  long  is  the  pole  ? 

9.  There  is  a  pole  two  fifths  under  water,  and  6 
feet  out  of  the  water ;  how  long  is  the  pole  ? 

10.  There  is  an  orchard,  in  which  3  sevenths  of  the 
trees  bear  cherries,  and   2  sevenths  bear  peaches, 
and  10  trees  bear  plums;  how  many  trees  are  there 
in  the  orchard  ?  and  how  many  of  each  sort  ? 

11.  There  is  a  school,  in  which  2  ninths  of  the 
boys  learn    arithmetic,  3  ninths  learn  grammar,  1 
ninth  learn  geography,  1  ninth  learn  geometry,  and 
12  learn  to  write ;  how  many  are  there  in  the  school  ? 
and  how  many  attending  to  each  study  ? 

12.  A  man  sold  a  watch  for  63  dollars,  which  was 
7  fifths  of  what  it  cost  him ;  how  much  did  he  gain 
by  the  bargain  ? 


SECT.  6.]  ARITHMETIC.  77 


Miscellaneous  Examples. 

1.  IF  1  yard  of  cloth  cost  4  dollars,  what  will  5 
yards  cost? 

2.  A  man  bought  3  pounds  of  raisins,  at  7  cents 
a  pound,  and  16  oranges,  at  4  cents  apiece,  and  1 
pound  of  candles  for  16  cents ;  what  did  they  all 
come  to? 

3.  A   hoy  had  37  apples;   he  kept  five  himself, 
and  divided  the   rest  equally  among  four  compan- 
ions;  how  many  did  he  give  them  apiece? 

4.  Two  men  are  40  miles  apart,  and  both  travel- 
ling the  same  way  ;  the  hindermost  man  gains  upon 
the  other  5  miles  each  day  ;  in  how  many  days  will 
he  overtake  him  ? 

5.  Two  men  are  travelling  the  same  way;  one 
travels  at  the  rate  of  38  miles  a  day  ;  the  other,  who 
is  behind  the  former,  travels  44  miles ;  how  much 
does   he  gain  of  the   first  each  day?    and  in  how 
many  days  would  he  gain  60  miles  ? 

6.  A  fox  is  80  rods  before  a  greyhound,  and  is 
running   at  the  rate  of  27  rods  in  a  minute ;  the 
greyhound  is  following  at  the  rate  of  31  rods  in  a 
minute ;  in  how  many  minutes  will  the  greyhound 
overtake  the  fox  ? 

7.  If  two  yards  of  cloth  cost  6  dollars,  what  would 
4  yards  cost?    What  would  12  yards  cost? 

8.  If  8  sheep  cost  24  dollars,  what  would  3  cost? 

9.  If  4  tons  of  hay  will  keep  3  horses  through  the 
winter,  how  many  tons  will  keep  30  horses  the  same 
time  ? 

10.  If  a  man  spends  8  shillings  a  day,  how  many 
dollars  will  he  spend  in  a  week  ? 

11.  Bought  10  pieces  of  cloth,  each  containing  5 
yards,  for  100  dollars  ;  what  was  it  a  piece  ?  and 
..how  much  a  yard  ? 


78  ARITHMETIC.  [PART  1. 

12.  If  the  wages  of  12  weeks  come  to  60  dol- 
lars, what  is  that  a  month  ?  and  how  much  for  5 
weeks  ? 

13.  If  7  horses  eat  14  bushels  of  oats  in  1  week, 
how  many  bushels  would  15  horses  eat  in  the  same 
time  ? 

14.  If  3  horses  eat  8  bushels  of  oats  in  2  weeks, 
how  long  would  it  take  them  to  eat  40  bushels  ? 

15.  If  1  horse  eat  1  bushel  of  oats  in  3  days,  in 
how  many  days  would  4  horses  eat  36  bushels? 

16.  If  2 'men  spend  12  dollars  in  I  week,  how 
many  dollars  would  3  men,  at  the  same  rate,  spend 
in  5  weeks? 

17.  If  a  staff  3  feet  long  cast  a  shadow  of  2  feet 
at  12  o'clock,  what  is  the  length  of  a  pole  that  casts 
a  shadow  IS  foot  at  the  same  time  of  day  ? 

18.  If  47  gallons  of  water,  in  1  hour,  run  into  a 
cistern  containing  10S  gallons,  and  by  a  pipe  38 
gallons  run  out  in  an  hour,  how  much  remains  in  the 
cistern  in  an  hour  1  and  in  how  many  hours  will  the 
cistern  be  filled  ? 

19.  If  4  men  can  do  a  piece  of  work  in  8  days, 
how  many  men  would  it  take  to  do  the  same  work 
in  4  days? 

20.  If  6  men  can  do  a  piece  of  work  in  9  days, 
in  how  many  days  would  2  men  do  it? 

21.  If  8  men  can  do  a  piece  of  work  in  5  days, 
in  how  many  days  would  they  do  a  piece  of  work  4 
times  as  large? 

22.  If  7  men  can  do  a  piece  of  work  in  3  days, 
how  many  men  would  it  take  to  do  a  piece  of  work 
5  times  as  large  in  the  same  time  ? 

23.  If  8  men  can  do  a  piece  of  work  in  4  days, 
in  how  many  days  would  2  men  do  a  piece  of  work 
one  half  as  large  ? 

24.  A  man  bought  a  cask  of  wine  containing  63 
gallons,  3  sevenths  of  which  leaked  out ;  and  he 


SECT.  7.]  ARITHMETIC.  79 

sold  the  remainder  for  36  dollars;  how  much  per 
gallon  did  he  sell  it  for  ? 

25.  If  a  cask  of  wine  cost  48  dollars,  what  is  5 
eighths  of  it  worth  1 

26.  A  man  bought  7  oranges  for  6  cents  apiece, 
and  sold  them  all  for  54  cents ;  how  much  did  he 
gain  by  the  bargain  ? 

27.  A  man  bought  8  yards  of  cloth  for  56  dol- 
lars, and  sold  it  again  for  9  dollars  a  yard ;    how 
much  did  he  gain  by  the  bargain  ? 

28.  A  man  bought  8  barrels  of  flour  for  40  dol- 
lars ;   how  much  must  he  sell  it  at  per  barrel  to  gain 
16  dollars  ? 

29.  A  man  bought  five  firkins  of  butter  at  7  dol- 
lars a  firkin ;  how  must  he  sell  it  per  firkin  to  gain 
10  dollars  ? 

30.  A  man  gave  35  cents  for  his  breakfast,  which 
was  5  eighths  of  what  he  gave  for  his  dinner ;  what 
did  he  give  for  his  dinner? 

31.  A  ship's  crew  of  6  men  have  provision  for  3 
months ;  how  many  months  would  it  last  1  man  ? 

32.  A  ship's  crew  have  provision  sufficient  to  last 
1  man  27  months ;  how  long  would  it  last  9  men  ? 

33.  A  ship's  crew  have  provision  sufficient  to  last 
3  men  ten  months ;  how  long  would  it  last  5  men  ? 

34.  A  man  built  40  rods  of  wall  in  a  certain  time, 
another  man  can  build  9  rods  while  the  first  builds 
5 ;  how  much  would  he  build  in  the  same  time  ? 


SECTION   VII. 

If  the  combinations  in  this  section  should  be  found 
too  difficult,  they  may  be  omitted  until  reviewing  the 
book. 

A..  1.  A  MAN,  being  asked  the  age  of  his  eldest 


80  ARITHMETIC.  [PART! 

son,  answered,  that  his  youngest  son  was  six  years 
old,  and  that  2  thirds  of  the  youngest  son's  age  was 
just  1  fifth  of  the  eldest  son's  age.  Required  the 
age  of  the  eldest  son. 

2.  2  thirds  of  6  is  1  fifth  of  what  number  ? 

3.  A  man,  being  asked  how  many  sheep  he  had, 
said  that  he  had  them  in  two  pastures ;  in  one  pas- 
ture he  had  eight ;  and  that  3  fourths  of  these  was 
just  1  third  of  what  he  had  in  the  other.     How 
many  were  there  in  the  other  ? 

4.  3  fourths  of  8  is  1  third  of  what  number  ? 

5.  Two  boys  talking  of  their  ages,  one  said  he 
was  9  years  old.     Well,  said  the  other,  2  thirds  of 
your  age  is  exactly  3  fourths  of  my  age ;   now,  if  you 
will  tell  me  how  old  I  am,  I  will  give  you  as  many 
apples  as  I  am  years  old.     What  was  his  age  ? 

6.  2  thirds  of  9  is  3  fourths  of  what  number  ? 

7.  Two  boys  counting  their  money,  one  said  he 
had  ten  cents.     The  other  says,  4  fifths  of  your 
money  is  exactly  2  sevenths  of  mine ;  now,  if  you 
will  tell  how  many  I  have,  I  will  give  you  1  half  of 
them.     How  many  had  he  ? 

8.  4  fifths  of  10  is  2  sevenths  of  what  number  ? 

9.  5  sixths  of  12  is  2  thirds  of  what  number? 

10.  6  sevenths  of  14  is  4  ninths  of  what  number? 

11.  6  fifths  of  15  is  2  thirds  of  what  number? 

12.  7  ninths  of  18  is  2  fifths  of  what  number  ? 

13.  4  fifths  of  20  is  8  sevenths  of  what  number  ? 

14.  8  sevenths  of  21  is  6  tenths  of  what  number? 

15.  5  sixths  of  24  is  10  sevenths  of  how  many 
times  5? 

16.  3  sevenths  of  28  is  2  eighths  of  how  many 
times  7? 

17.  4  fifths  of  30  is  6  sevenths  of  how  many 
times  8  ? 

18.  6  eighths  of  32  is  8  ninths  of  how  many 
times  5  ? 


SECT.  7.]  ARITHMETIC.  81 

19.  4  ninths  of  36  is  8  tenths  of  how  many  times 
6? 

20.  3  fourths  of  40  is  5  sevenths  of  how  many 
times  8  ? 

21.  6  ninths  of  45  is  3  fifths  of  how  many  times 
7? 

22.  5  sixths  of  48  is  10  sevenths  of  how  many 
times  3  ? 

23.  4  sevenths  of  63  is  6  fifths  of  how  many 
times  8  ? 

24.  5  ninths  of  72  is  4  sevenths  of  how  many 
times  9  ? 

B.  1.  4  fifths  of  15  is  6  tenths  of  how  many 
thirds  of  21  ? 

2.  4  thirds  of  IS  is  8  ninths  of  how  many  sevenths 
of  35? 

3.  6  sevenths  of  21  is  2  thirds  of  how  many  thirds 
of  24? 

4.  5  fourths  of  24  is  10  sevenths  of  how  many 
fifths  of  40  ? 

5.  5  eighths  of  32  is  2  fifths  of  how  many  fifths 
of  35? 

6.  4  sevenths  of  63  is  6  eighths  of  how  many 
ninths  of  45  ? 

7.  3  sevenths  of  56  is  4  ninths  of  how  many 
fourths  of  28  ? 

8.  3  eighths  of  64  is  6  tenths  of  how  many  sixths 
of  30? 

9.  2  eighths  of  72  is  3  tenths  of  how  many  fifths 
of  40? 

C.  1.  Two  times  eleven  are  how  many? 

2.  Two  times  twelve  are  how  many  ? 

3.  Two  times  thirteen  are  how  many  ? 

4.  Two  times  fourteen  are  how  many  ? 

5.  Two  times  fifteen  are  how  many  ? 


82  ARITHMETIC.  [PART  1. 

6.  Two  times  sixteen  are  how  many  ? 

7.  Two  times  seventeen  are  how  many  1 

8.  Two  times  eighteen  are  how  many  ? 

9.  Two  times  nineteen  are  how  many  ? 

10.  Two  times  twenty  are  how  many? 

11.  Three  times  eleven  are  how  many  ? 

12.  Three  times  twelve  are  how  many  ? 

13.  Three  times  thirteen  are  how  many  ? 

14.  Three  times  fourteen  are  how  many  ? 

15.  Three  times  fifteen  are  how  many  ? 

16.  Three  times  sixteen  are  how  many? 

17.  Three  times  seventeen  are  how  many  ? 

18.  Three  times  eighteen  are  how  many? 

19.  Three  times  nineteen  are  how  many  ? 

20.  Three  times  twenty  are  how  many  ? 

21.  Four  times  eleven  are  how  many? 

22.  Four  times  twelve  are  how  many  ?  ' 

23.  Four  times  thirteen  are  how  many  ? 

24.  Four  times  fourteen  are  how  many  ? 

25.  Four  times  fifteen  are  how  many  ? 

26.  Four  times  sixteen  are  how  many? 

27.  Four  times  seventeen  are  how  many  ? 

28.  Four  times  eighteen  are  how  many  ? 

29.  Four  times  nineteen  are  how  many? 

30.  Four  times  twenty  are  how  many  ? 

31.  Five  times  eleven  are  how  many  1 

32.  Five  times  twelve  are  how  many  ? 

33.  Five  times  thirteen  are  how  many  ? 

34.  Five  times  fourteen  are  how  many  ? 

35.  Five  times  fifteen  are  how  many  ? 

36.  Five  times  sixteen  are  how  many  ? 

37.  Five  times  seventeen  are  how  many  ? 

38.  Five  times  eighteen  are  how  many  ? 

39.  Five  times  nineteen  are  how  many  ? 

40.  Five  times  twenty  are  how  many? 

41.  Six  times  eleven  are  how  many? 

42.  Six  times  twelve  are  how  many  ? 


SECT.  7.]  ARITHMETIC. 

43.  Six  times  thirteen  are  how  many  1 

44.  Six  times  fourteen  are  how  many  1 

45.  Six  times  fifteen  are  how  many  ? 

46.  Six  times  sixteen  are  how  many  ? 

47.  Six  times  seventeen  are  how  many  ? 

48.  Six  times  eighteen  are  how  many  ? 

49.  Six  times  nineteen  are  how  many  ? 

50.  Six  times  twenty  are  how  many  ? 

51.  Seven  times  eleven  are  how  many? 

52.  Seven  times  twelve  are  how  many  ? 

53.  Seven  times  thirteen  are  how  many  ? 

54.  Seven  times  fourteen  are  how  many? 

55.  Seven  times  fifteen  are  how  many  ? 

56.  Seven  times  sixteen  are  how  many  ? 

57.  Seven  times  seventeen  are  how  many  ? 

58.  Seven  times  eighteen  are  how  many  ? 

59.  Seven  times  nineteen  are  how  many  1 

60.  Seven  times  twenty  are  how  many  1 

61.  Eight  times  eleven  are  how  many  ? 

62.  Eight  times  twelve  are  how  many  ? 

63.  Eight  times  thirteen  are  how  many  ? 

64.  Eight  times  fourteen  are  how  many  ? 

65.  Eight  times  fifteen  are  how  many  ? 

66.  Eight  times  sixteen  are  how  many  ? 

67.  Eight  times  seventeen  are  how  many  ? 

68.  Eight  times  eighteen  are  how  many  ? 

69.  Eight  times  nineteen  are  how  many  1 

70.  Eight  times  twenty  are  how  many  ? 

71.  Nine  times  eleven  are  how  many  1 

72.  Nine  times  twelve  are  how  many? 

73.  Nine  times  thirteen  are  how  many  ? 

74.  Nine  times  fourteen  are  how  many  ? 

75.  Nine  times  fifteen  are  how  many  ? 

76.  Nine  times  sixteen  are  how  many  ? 

77.  Nine  times  seventeen  are  how  many  ? 

78.  Nine  times  eighteen  are  how  many  ? 

79.  Nine  times  nineteen  are  how  many? 


84  ARITHMETIC.  [PART  1. 

80.  Nine  times  twenty  are  how  many  ? 

81.  Ten  times  eleven  are  how  many  1 

82.  Ten  times  twelve  are  how  many  ? 

83.  Ten  times  thirteen  are  how  many  ? 

84.  Ten  times  fourteen  are  how  many? 

85.  Ten  times  fifteen  are  how  many  ? 

86.  Ten  times  sixteen  are  how  many  ? 

87.  Ten  times  seventeen  are  how  many  ? 

88.  Ten  times  eighteen  are  how  many  ? 

89.  Ten  times  nineteen  are  how  many  ? 

90.  Ten  times  twenty  are  how  many  ? 


SECTION  VIII. 

A.   1.  IF  you  cut  an  apple  into  two  equal  parts, 
what  is  one  of  those  parts  called?* 

2.  How  many  halves  of  an  apple  will  make  4he 
whole  apple? 

3.  If  you  cut  an  apple  into  3  equal  paits,  what  is 
1  of  those  parts  called?     What  are  2  of  the  parts 
called  ? 

4.  How  many  thirds  of  an  apple  will  make  the 
whole  apple? 

5.  If  you  cut  an  apple  into  4  equal  parts,  what  is 
1  of  those  parts  called  ?  What  are  2  of  those  parts 
called  ?     What  are  three  of  them  called  ? 

6.  How   many    fourths   of    an    apple    make   the 
whole  apple  ? 

7.  If  an  apple  be  cut  into  5  equal  parts,  what 
is  one  of  the  parts  called  ?     What  are  2  of  the  parts 
called  ?    What  are  3  of  the  parts  called  ?    What  are 
4  of  the  parts  called  ? 

*  See  Sect.  III.  Art.  B.,  remark  before  question  1  and  17. 


SECT.  8.]  ARITHMETIC.  85 

8.  How  many  fifths  of  an  apple  make  the  whole 
apple  ? 

9.  If  an  apple  be  cut  into  6  equal  parts,  what  is 
1  of  the*  parts  called?     What  are  2  of  the  parts 
called?    What  3?    What  4?    What  5? 

10.  How  many  sixths  of  an  apple  make  the  whole 
apple  ?     , 

11.  If  an  apple  be  cut  into  7  equal  parts,  what  is 
1  of  the  parts  called  ?    What   are  2  of  the  parts 
called?    What  3?    What  4?    What  5?    What  6? 

Let  the  instructer  ask  the  pupil  the  divisions  of  a 
unit  in  this  manner  as  far  as  the  division  into  10  parts. 
It  would  be  well  to  ask  them  further.  Then  let  him 
begin  again,  and  suppose  an  orange  instead  of  an 
apple.  After  applying  the  division  to  several  differ- 
ent things,  Plate  II.  may  be  explained  and  used. 
It  will  often  be  found  useful  to  refer  the  pupil  to  the 
divisions  of  some  sensible  object.  For  the  explana- 
tion of  Plate  II.  see  the  Key. 

12.  A  man  had  a  bushel  of  corn,  and  wished  to 
give  1  half  of  a   bushel   apiece  to  some  laborers ; 
how  many  could  he  give  it  to  ? 

13.  How  many  halves  are  there  in  1  ? 

14.  A  man  divided  2  barrels  of  flour  among  his 
laborers,  giving   them  1  half  of  a  barrel    apiece; 
how  many  men  did  he  give  it  to  ? 

15.  How  many  halves  are  there  in  2  ?* 

16.  In  3  bushels  of  corn  how  many  half  bushels  ? 

17.  How  many  halves  are  there  in  3? 

18.  A  boy  divided  4  oranges  among  his  compan- 
ions, giving  them  1  half  of  an  orange  apiece;  how 
many  boys  did  he  give  them  to  ? 

19.  How  many  halves  are  there  in  4  ? 

*  Be  careful  to  make  the  pupil  use  the  plate.  He  might  answer  the 
questions  without,  but  he  will  not  understand  their  meaning  so  welK 


86  ARITHMETIC.  [PART  1. 

20.  A  man,  having  some  laborers,  gave  them  1 
half  a  dollar  apiece  ;  it  took  3  dollars  and  1  half  a 
dollar    to    pay   them ;    how    many   laborers    were 
there? 

21.  How  many  halves  are  there  in  3  and  1  half? 

22.  How  many  halves  are  there  in  5  ? 

23.  How  many  halves  are  there  in  7  and  1  half? 

24.  How  can  you  tell  how  many  halves  there  are 
in  any  number? 

Answer.  Since  there  are  two  halves  in  one,  there 
will  be  twice  as  many  halves  as  there  are  whole 
ones. 

25.  If  you   had  1  orange,   and  should  divide  it 
among  your  companions,  giving  them  1  third  apiece, 
how  many  could  you  give  it  to  ? 

26.  How  many  thirds  are  there  in  1  ? 

27.  If  you  cut  2  oranges  each  into  3  pieces,  how 
many  pieces  would  they  make  ? 

28.  If  you  cut  3  oranges  into  3  pieces  each,  how 
many  pieces  would  they  make  ? 

29.  If  you  cut  4  apples  each  into  3  pieces,  how 
many  pieces  would  they  make  ? 

30.  How  many  thirds  are  there  in  2  ?  In  3  ?  In  4? 
In  5? 

31.  How  can  you  tell  how  many  thirds  there  are 
in  any  number? 

Answer.  Since  there  are  3  thirds  in  one,  there  will 
be  3  times  as  many  thirds  as  there  are  whole  ones. 

32.  If  you  had  2  bushels  and  1  third  of  a  bushel 
of  corn  to  give  to  some  poor  persons,  how  many 
could  you  give  it  to  if  you  should  give  them  1  third 
of  a  bushel  apiece  ? 

33.  How  many  thirds  are  there  in  2  and  1  third  ? 

34.  If  a  horse  can  eat  1  third  of  a  bushel  of  oats 
in  1  day,  how  many  days  would  it  take  him  to  eat 
3  bushels  and  2  thirds  of  a  bushel  ? 

35.  How  many  thirds  are  there  in  3  and  2  thirds  ? 


SECT.  8.]  ARITHMETIC.  87 

36.  If  1  horse  can  eat  1  third  of  a  bushel  of  oats 
in  a  day,  how  many  horses  will  it  take  to  eat  5  bush- 
els and  2  thirds  of  a  bushel  in  the  same  time? 

37.  In  5  and  2  thirds  how  many  thirds  ? 

38.  In  seven  and  1  third  hgw  many  thirds  ? 

39.  If  1  horse  will  eat  1  fourth  of  a  ton  of  hay  in 
1  month,  how  many  horses  will  eat  a  ton  in  the  same 
time  ?    How  many  will  eat  2  tons  ?    How  many  will 
eat  3  tons  ?  4  tons  1  5  tons  ?  6  tons  ? 

40.  How  many  fourths  are  there  in  1?   In  2?  In 
3?  In  4?  In  5?  In  6? 

41.  How  can  you  tell  how  many  fourths  there  are 
in  any  number  ? 

42.  How  many  fourths  are  there  in  4  and  1  fourth? 

43.  In  5  tons  of  hay  and  3  fourths  of  a  ton,  how 
many  fourths  of  a  ton  1 

44.  In  5  and  3  fourths  how  many  fourths  ? 

45.  In  7  and  1  fourth  how  many  fourths  ? 

46.  In  9  and  3  fourths  how  many  fourths  ? 

47.  If  a  horse  eat  1  fifth  of  a  ton  of  hay  in  a  month, 
how  many  horses  will  eat  a  ton  in  the  same  time  ?  2 
tons  ?  3  tons  ?  i  tons  ?  5  tons  ? 

48.  How  many  fifths  are  there  in  1?   In  2?  In  3? 
In  4?  In  5?  In  7?  In  9? 

49.  How  can  you  tell  how  many  fifths  there  are 
in  any  number  ? 

50.  In  2  and  1  fifth  how  many  fifths? 

51.  In  3  dollars  and  2  fifths  of  a  dollar  how  many 
fifths  of  a  dollar  ? 

52.  In  3  and  2  fifths  how  many  fifths  ? 

53.  In  5  and  3  fifths  how  many  fifths  ? 

54.  In  6  and  4  fifths  how  many  fifths  ? 

55.  How  many  sixths  are  there  in  1?  In  2?  In 
3?  In  4?  In  5?  In  7?  In  8? 

56.  In  2  and  2  sixths  how  many  sixths  ? 

57.  In  3  and  4  sixths  how  many  sixths  ? 

58.  In  6  and  5  sixths  how  many  sixths  ? 


88  ARITHMETIC.  [PART  1. 

59.  How  many  sevenths  are  there  In  1  ?    In  2  ? 
In  3?  In  4?  In  6?  In  9?  In  10? 

60.  In  3  and  1  seventh  how  many  sevenths? 

61.  In  5  and  3  sevenths  how  many  sevenths? 

62.  In  7  and  5  sevenths  how  many  sevenths  ? 

63.  How  many  eighths  are  there  in  1  ?    In  3?  In 
5?  In8? 

64.  In  2  and  3  eighths  how  many  eighths  ? 

65.  In  3  and  5  eighths  how  many  eighths  ? 

66.  In  5  and  7  eighths  how  many  eighths? 

67.  How  many  ninths  are  there  in  1  ?  In  2  ?  In 
7?  In5? 

68.  In  2  and  2  ninths  how  many  ninths  ? 

69.  In  4  and  3  ninths  how  many  ninths  ? 

70.  In  6  and  4  ninths  how  many  ninths  ? 

71.  In  8  and  7  ninths  how  many  ninths? 

72.  How  many  tenths  in  1  ?  In  2  ?  In  5?  In  8? 

73.  In  3  and  3  tenths  how  many  tenths  ? 

74.  In  4  and  7  tenths  how  many  tenths? 

75.  In  8  and  9  tenths  how  many  tenths? 

76.  In  7  and  4  tenths  how  many  tenths  ? 

77.  In  9  and  8  tenths  how  many  tenths  ? 

78.  In  7  and  4  sevenths  how  many  sevenths  ? 

79.  In  9  and  2  thirds  how  many  thirds  ? 

80.  In  10  and  3  fourths  how  many  fourths? 

81.  In  8  and  4  fifths  how  many  fifths  ? 

82.  In  7  and  5  ninths  how  many  ninths? 

B.  1.  If  you  give  4  men  1  half  of  a  barrel  of  flour 
apiece,  how  many  barrels  will  it  take  ? 

2.  In  4  halves  how  many  times  1  ? 

3.  If  you  give  3  boys  1  half  of  an  orange  apiece, 
how  many  oranges  will  it  take  ? 

4.  In  3  halves  how  many  times  1  ? 

5.  If  you  give  five  men  1  half  of  a  dollar  apiece, 
how  many  dollars  will  it  take  ? 

6.  In  5  halves  how  many  times  1  ? 


SECT  8.]  ARITHMETIC.  89 

7.  In  6  halves  how  many  times  1  ? 

8.  In  7  halves  how  many  times  1  ? 

9.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  halves  ? 

10.  A  man  divided  some  corn  among  6  persons, 
giving  them  1  third  of  a  bushel  apiece ;  how  many 
bushels  did  it  take  ? 

11.  In  6  thirds  how  many  times  1  ? 

12.  In  5  thirds  how  many  times  1  ? 

13.  A  man  gave  eight  paupers  1  third  of  a  dollar 
apiece  ;  how  many  dollars  did  it  take  ? 

14.  In  8  thirds  how  many  times  1  ? 

15.  In  10  thirds  how  many  times  1  ? 

16.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  thirds  ? 

17.  If  a  man  spends  1  fourth  of  a  dollar  in  one 
day,  how  many  dollars  will  he  spend  in  8  days? 
How  many  in  7  days?  How  many  in  11  days  ? 

18.  In  8  fourths  how  many  times  1  ? 

19.  In  7  fourths  how  many  times  1  ? 

20.  In  11  fourths  how  many  times  1  ? 

21.  In  13  fourths  how  many  times  1  ? 

22.  In  18  fourths  how  many  times  1  ? 

23.  How  can  you  tell  how  many  whole  ones  there 
are  in  any  number  of  fourths  ? 

24.  If  1  fifth  of  a  barrel  of  beer  will  last  a  family 
1  day,  how  many  barrels  will  last  them  10  days? 
How  many  8  days?  11  days ?  17  days ? 

25.  In  10  fifths  how  many  times  1  ? 

26.  In  8  fifths  how  many  times  1  ? 

27.  In  1 1  fifths  how  many  times  1  ? 

28.  In  17  fifths  how  many  times  1? 

29.  In  18  sixths  how  many  times  1  ? 

30.  In  23  fifths  how  many  times  1  ? 

31.  In  21  sevenths  how  many  times  1  ? 

32.  In  24  eighths  how  many  times  1  ? 

33.  In  36  ninths  how  many  times  1  ? 

8* 


90  ARITHMETIC.  [!»ABT  1. 

34.  In  30  tenths  how  many  times  1  ? 

35.  In  35  fourths  how  many  times  1  ? 

36.  In  37  eighths  how  many  times  1  ? 

37.  In  43  fifths  how  many  times  1  ? 

38.  In  48  ninths  how  many  times  1  ? 

39.  In  53  tenths  how  many  times  1  ? 

40.  In  57  eighths  how  many  times  1  ? 

41.  In  76  tenths  how  many  times  1  ? 

42.  In  78  ninths  how  many  times  1 1 


SECTION  IX. 

A..   1.  IF  a  breakfast  for  1  man  cost  1  third  of  a 
dollar,  what  would  a  breakfast  for  two  men  cost  ? 

2.  How  much  is  2  times  1  third  ? 

3.  If  it  take  you  I  third  of  an  hour  to  travel  1 
mile,  how  long  will  it  take  you  to  travel  3  miles  ? 

4.  How  much  is  3  times  1  third  ? 

5.  If  1  man  can  eat  1  third  of  a  pound  of  meat  at 
a  meal,  how  much  can  5  men  eat? 

6.  How -much  is  7  times  1  third? 

7.  If  1  man  can  eat  2  thirds  of  a  pound  of  meat 
for  dinner,  how  many  thirds  of  a  pound  would  3 
men  eat? 

8.  How  much  is  2  times  2  thirds  ? 

9.  A  man  gave  to  4  paupers  2  thirds  of  a  dollar 
apiece ;    how  many  thirds  of  a  dollar  did   he  give 
them  ?    How  many  dollars  ? 

10.  5  times  2  thirds  are  how  many  thirds  ?  How 
many  times  1  ? 

11.  If  you  give  3  men  1  fourth  of  a  dollar  apiece, 
how  many  fourths  of  a  dollar  will  it  take  ? 

12.  3  times  1  fourth  are  how  many  fourths  ? 

13.  If  you  give  3  men  3  fourths  of  a  bushel  of 


SECT.  9.]  ARITHMETIC.  91 

corn  apiece,  how  many  fourths  of  a  bushel  will  it 
take  ?     How  many  bushels  ? 

14.  5  times  3  fourths  are  how  many  fourths  ?   How 
many  times  1? 

15.  If  1  horse  eat  1  fifth  of  a  bushel  of  oats  in  a 
day,  how  much  will  4  horses  eat  in  the  same  time  ? 

16.  3  times  1  fifth  are  how  many  fifths? 

17.  If  1  man  can  earn  3  fifths  of  a  dollar  in  a  day, 
how  much  can  he  earn  in  4  days  ? 

18.  7  times  3  fifths  are  how  many  fifths?    How 
many  times  1? 

19.  If  a  family  consume  2  sevenths  of  a  barrel  of 
flour  in  a  week,  how  much  would  they  consume  in 
5  weeks  ? 

20.  6  times  2  sevenths  are  how  many  sevenths? 
How  many  times  1? 

21.  5  times  3  eighths   are  how  many  eighths? 
How  many  times  1  ? 

22.  How  much  is  6  times  3  fifths  ? 

23.  How  much  is  7  times  5  sixths  ? 

24.  How  much  is  5  times  4  ninths  ? 

25.  How  much  is  6  times  8  ninths? 

26.  How  much  is  7  times  9  tenths  ? 

27.  How  much  is  5  times  7  tenths  ? 

28.  How  much  is  6  times  7  eighths  ? 

29.  How  much  is  9  times  5  eighths  ? 

30.  How  much  is  8  times  5  sevenths  ? 

31.  How  much  is  7  times  5  sixths? 
32. .  How  much  is  8  times  7  fourths  ? 

33.  How  much  is  7  times  4  fifths  ? 

34.  How  much  is  5  times  3  eighths  ? 

IS.  1.  If  1  bushel  of  wheat  cost  a  dollar  and  1 
half,  what  will  two  bushels  cost  ? 

2.  How  much  is  2  times  1  and  1  half?* 

*  This  is  to  be  understood,  2  times  1  and  2  times  1  half,  and  to 
be  answered  thus :  2  times  1  are  2;  and  2  times  1  half  are  2  halves 
or  1,  which,  added  to  2;  makes  3, 


92  ARITHMETIC.  [PART  1. 

3.  If  a  barrel  of  cider  cost  2  dollars  and  a  half, 
what  will  3  barrels  cost  ? 

4.  How  much  is  4  times  2  and  1  half  ? 

5.  If  a  barrel  of  beer  cost  3  dollars  and  a  half, 
what  will  2  barrels  cost  ? 

6.  How  much  is  5  times  3  and  1  half  1 

7.  How  much  is  6  times  3  and  1  half? 

8.  If  a  box  of  butter  cost  2  dollars  and  1  third  of 
a  dollar,  what  will  3  boxes  cost  ? 

9.  How  much  is  4  times  2  and  1  third? 

10.  If  you  give  to  two  persons  3  bushels  and  1 
third  of  a  bushel  of  wheat  apiece,  how  many  bushels 
will  it  take  ? 

11.  How  much  is  5  times  3  and  1  third? 

12.  If  you  give  to  4  persons  each  2  oranges  and  1 
fourth  of  an  orange,  how  many  oranges  will  it  take? 

13.  How  much  is  5  times  2  and  1  fourth  ? 

14.  If  it  take  3  yards  and  2  thirds  of  a  yard  of 
cloth  to  make  a  suit  of  clothes,  how  many  yards  will 
it  take  to  make  2  suits  ? 

15.  How  much  is  4  times  3  and  2  thirds  ? 

16.  If  a  family  consume  2  bushels  and  2  thirds  of 
a  bushel  of  malt  in  1  month,  how  much  will  they 
consume  in  3  months  ? 

17.  How  much  is  5  times  2  and  2  thirds  ? 

18.  How  much  is  4  times  3  and  3  fourths  ? 

19.  How  much  is  2  times  3  and  I  fourth? 

20.  How  much  is  3  times  3  and  3  fourths  ? 

21.  How  much  is  3  times  5  and  I  fourth  ? 

22.  If  a  horse  eat  3  tons  and  1  fifth  of  a  ton  of 
hay  in  a  year,  how  much  will  2  horses  eat  in  the 
same  time  ? 

23.  How  much  is  4  times  3  and  1  fifth  ? 

24.  If  a  man  can  travel  4  miles  and  2  fifths  of  a 
mile  in  one  hour,  how  far  will  he  travel  in  3  hours? 

25.  How  much  is  5  times  4  and  2  fifths  ? 

26.  How  much  is  3  times  5  and  3  fifths  ? 


SECT.  9.]  ARITHMETIC.  93 

27.  How  much  is  4  times  6  and  %  fifths  1 

28.  How  much  is  3  times  2  and  1  sixth  1 

29.  How  much  is  2  times  5  and  3  sixths  ? 

30.  If  a  yard  of  cloth  cost  4  dollars  and  5  sixths 
of  a  dollar,  what  will  4  yards  cost  ? 

31.  How  much  is  7  times  4  and  5  sixths  1 

32.  How  much  is  2  times  3  and  3  sevenths  ? 

33.  How  much  is  3  times  4  and  3  sevenths  1 

34.  If  a  bushel  of  wheat  cost  7  shillings  and  3 
eighths,  what  will  5  bushels  cost  at  that  rate  ? 

35.  How  much  is  3  times  4  and  5  eighths  1 

36.  How  much  is  4  times  3  and  7  eighths  1 

37.  A  man  bought  8  yards  of  cloth,  at  9  dollars 
and  3  tenths  a  yard ;  how  much  did  it  come  to  ? 

38.  How  much  is  6  times  2  and  2  ninths  ? 

39.  How  much  is  4  times  5  and  3  ninths  1 

40.  A  man  bought  10  barrels  of  cider  at  3  dollars 
and  5  sixths  a  barrel ;  how  much  did  it  come  to  ? 

41.  How  much  is  3  times  7  and  2  tenths  1 

42.  What  cost  8  barrels  of  beef,  at  9  dollars  and 
3  sevenths  a  barrel? 

C.  1.  A  boy  wished  to  give  5  other  boys  1  half 
pint  of  chestnuts  apiece ;  how  many  pints  would  it 
take? 

2.  A  boy  wished  to  give  3  other  boys  3  fourths  of 
an  orange  apiece ;  how  many  oranges  would  it  take  ? 

3.  A  man  gave  to  10  persons  1  fifth  of  a  bushel 
of  corn  apiece ;  how  many  bushels  did  it  take  ? 

4.  A  man  gave  to  7  men  3  fourths  of  a  gallon  of 
beer  apiece ;  how  many  gallons  of  beer  did  it  take  ? 

5.  If  it  take  1  yard  and  1  fourth  of  a  yard  of  cloth 
to  make  a  pair  of  pantaloons,  how  many  yards  would 
it  take  to  make  8  pairs  ? 

6.  If  a  family  consume  2  bushels  and  3  fifths  of  a 
bushel  of  grain  in  1  week,  how  many  bushels  would 
they  consume  in  4  weeks  ? 


94  ARITHMETIC.  [PART  1. 

7.  If  a  horse  eat  3  bushels  and  4  sevenths  of  a 
bushel  of  oats  in  1  week,  how  many  bushels  would 
he  eat  in  8  weeks  1 

8.  If  a  horse  eat  5  loads  and  3  eighths  of  a  load  of 
hay  in  1  year,  how  many  loads  would  6  horses  eat  ? 

9.  If  a  man  travel  4  miles  and  5  ninths  in  an 
hour,  how  many  miles  would  he  travel  in  8  hours  ? 

10.  If,  in  an  orchard  of  10  trees,  each  tree  bears 
8  bushels  and  3  sevenths,  how  many  bushels  will 
the  whole  orchard  bear  1 

11.  If  a  man  can  build  5  rods  and  7  eighths  of  a 
rod  of  wall  in  1  day,  how  many  rods  can  he  build 
in  8  days  ? 

12.  If  3  men  can  build  a  piece  of  wall  in  4  days 
and  3  fifths  of  a  day,  how  many  days  would  it  take 
1  man  to  build  it? 

13.  If  1  man  can  build  7  rods  and  2  fifths  ef  wall 
in  a  day,  how  many  rods  would  10  men  build  ? 

14.  If  1  man  build  3  rods  and  2  ninths  of  wall  in 
1  day,  how  many  rods  would  3  men  build  in  4  days  ? 

15.  If  it  take  1  yard  and  3  sevenths  of  a  yard  of 
cloth  to  make  1  pair  of  pantaloons,  and  2   yards 
and  4  sevenths  for  a  coat,  how  many  yards  would 
it  take  to  make  3  pairs  of  pantaloons  and  3  coats? 


SECTION  X. 

A.  1.  IF  a  yard  of  cloth  cost  3  dollars,  what  will 
1  half  of  a  yard  cost  ? 

2.  What  is  1  half  of  3?* 

3.  If  a  barrel  of  beer  cost  5  dollars,  what  will  1 
lialf  of  a  barrel  cost? 

*  See  Sect.  V.  Art.  A. 


SECT.  10.]  ARITHMETIC.  95 

4.  What  is  1  half  of  5? 

5.  If  2  barrels  of  cider  cost  7  dollars,  what  is 
that  a  barrel  ? 

6.  What  is  1  half  of  7? 

7.  What  is  1  half  of  9? 

8.  What  is  1  half  of  11? 

9.  What  is  1  half  of  13? 

10.  What  is  1  half  of  15? 

11.  If   you    divide  1  bushel    of    wheat    equally 
among  3  persons,  what  part  of  a  bushel  will  you 
give  them  apiece? 

12.  If  3  yards  of  cloth  cost  1  dollar,  what  part 
of  a  dollar  will  1  yard  cost? 

13.  What  is  1  third  of  1? 

14.  How  could  you  divide  2  oranges  into  3  equal 
parts?     that   is,   how   can   you   find  1  third   of  2 
oranges  ?  * 

15.  One  third  of  2  oranges  will  be  the  same  as 
how  many  thirds  of  one  orange  ? 

16.  If   you   divide  2   bushels  of  wheat   equally 
among  3  persons,  what  part  of  a  bushel  will  you 
give  them  apiece? 

17.  If  3  bushels  of  corn  cost  2  dollars,  what  part 
of  a  dollar  will  1  bushel  cost  ? 

NOTE.  One  third  of  two  things  is  twice  as  much  as 
one  third  of  one  thing.  One  third  of  one  is  one  third, 
and  consequently  one  third  of  two  things  is  two  thirds. 
In  the  same  manner,  one  third  of  four  things  is  four 
thirds  of  one  thing.  If  four  oranges  be  cut  each  into 
three  parts,  and  then  one  part  of  each  be  taken,  it  will 
make  four  pieces,  each  of  which  is  one  third  of  one 
orange.  Hence  one  third  of  four  oranges  is  four  thirds 
of  one  orange,  that  is,  one  whole  one  and  one  third. 

18.  If  3  bushels  of  wheat  cost  4   dollars,  how 
much  is  that  a  bushel? 

*  Divide  each  orange  into  three  parts,  and  then  take  one  part  from 
each. 


96  ARITHMETIC.  [PART  1. 

19.  What  is  one  third  of  2  ?    Of  4  ? 

20.  If  3  gallons  of  wine  cost  5  dollars,  what  is 
that  a  gallon  ? 

21.  What  is  1  third  of  5?  Of  71   Of  8?  Of  10? 
Of  11? 

22.  If   a  bushel   of   apples   be    divided   equally 
among  4  persons,  what  part  of  a  bushel  will  they 
have  apiece?    What  would  they  have  apiece,  if  2 
bushels  were  divided  among  them  ?   What,  if  3  bush- 
els?   What,  if  5  bushels?    What,  if  6  bushels ? 

23.  What  is  1  fourth  of  1?   Of  2?   Of  3?   Of  5? 
Of  6?    Of  7?   Of  9?   Of  10? 

24.  If  a  bushel  of  malt  will  serve  5  persons  1 
month,  how  much  will  serve  1  person  the  same  time  ? 

25.  If  2  barrels  of  cider  will  serve  5  persons  1 
month,    how   much   will    serve  1  person   the   same 
time? 

26.  If  3  barrels  of  flour  be  divided  among  5  men, 
how  much  will  each  have?  If  4  barrels  were  divided, 
what  would  each  have?    What,  if  6  barrels  were 
divided  ?    What,  if  7  barrels  were  divided  ? 

27.  What  is  1  fifth  of  1?    Of  2?    Of  3?   Of  4? 
Of  6?    Of  7? 

28.  What  is  1  sixth  of  1  ?    Of  2?    Of  3?    Of  4? 
Of  5?   Of  7?   Of  8?    Of  9?   Of  10? 

29.  What  is  1  seventh  of  1?    Of  2?    Of  3?    Of 
4?    Of  5?    Of  6? 

30.  What  is  1  eighth  of  1?   Of  2?   Of  3?   Of  4? 
Of  5?   Of  6?   Of  7?   Of  8?   Of  9?   Of  10? 

31.  What  is  1  ninth  of  1?    Of  2  ?    Of  3?    Of  4? 
Of5?   Of6?   Of7?   Of8?  Of9?  OflO?  Ofll? 

32.  What  is  1  tenth  of  1?    Of  2?    Of  3?    Of  4? 
Of  5?   Of  6?    Of  7?   Of  8?   Of  9?    OflO?  Of 
11?   Of  12?   Of  13? 

33.  If  3  yards  of  cloth  cost  2  dollars,  what  will 
1  yard  cost?    What  will  2  yards  cost  ? 

34.  If  1  bushel  of  wheat  cost  2   dollars,  what 


SECT.  10.]  ARITHMETIC.  97 

will  1  third  of  a  bushel  cost?  What  will  2  thirds  of 
a  bushel  cost? 

35.  What  is  1  third  of  2  ? 

36.  What  is  2  thirds  of  2  ? 

37.  If  a  load  of  wood  cost  5  dollars,  what  will  1 
third  of  a  load  cost  ?    What  will  2  thirds  of  a  load 
cost? 

38.  What  is  1  third  of  5  ? 

39.  What  is  2  thirds  of  5  ? 

40.  What  is  2  thirds  of  7  ? 

41.  If  4  bushels  of  salt  cost  3  dollars,  what  will  1 
bushel  cost  ?  What  will  3  bushels  cost  ? 

42.  What  is  1  fourth  of  3? 

43.  What  is  3  fourths  of  3  ? 

44.  If  a  barrel  of  cider  cost  2  dollars,  what  will 
1  fifth   of  a  barrel  cost?  What  will  2  fifths  of  a 
barrel  cost  ? 

45.  What  is  1  fifth  of  2? 

46.  What  is  2  fifths  of  2? 

47.  What  is  one  fifth  of  7  ? 

48.  What  is  3  fifths  of  7? 

49.  If  7  gallons  of  gin  cost  5  dollars,  what  will  1 
gallon  cost  ?  What  will  4  gallons  cost  ? 

50.  What  is  1  seventh  of  5  ? 

51.  What  is  4  sevenths  of  5 1 

52.  What  is  1  sixth  of  4? 

53.  What  is  5  sixths  of  4  ? 

54.  If  you  divide  7  dollars  among  8  men,  what 
part  of  a  dollar  will  you  give  them  apiece  ?    What 
would  three  of  them  have  ? 

55.  What  is  1  eighth  of  7  ? 

56.  What  is  3  eighths  of  7? 

57.  What  is  1  ninth  of  10? 

58.  What  is  7  ninths  of  10? 

59.  What  is  1  tenth  of  14  ? 

60.  What  is  4  tenths  of  14  ? 

61.  If  5  yards  of  cloth  cost  17  dollars,  what  is 

9 


98  ARITHMETIC.  [PART!. 

that  a  yard  ?  What  would  3  yards  cost  ?  What  would 
8  yards  cost  ? 

<IW.  What  is  3  fifths  of  17? 

63.  What  is  8  fifths  of  17? 

<>4.  What  is  5  sevenths  of  20  ? 

65.  What  is  2  ninths  of  22  ? 

66.  What  is  3  eighths  of  27? 

67.  What  is  2  thirds  of  28  ? 

68.  What  is  3  fourths  of  31  ? 

69.  If  5  loads  of  hay  cost  47  dollars,  what  is  that 
a  load  ?  What  will  2  loads  cost  ?    What  will  7  loads 
cost  ?  What  will  12  loads  cost  ? 

70.  What  is  2  fifths  of  47? 

71.  What  is  7  fifths  of  47? 

72.  What  is  12  fifths  of  47  ? 

73.  What  is  4  sevenths  of  48? 

74.  What  is  4  ninths  of  50  ? 

75.  What  is  2  sevenths  of  58? 

76.  What  is  3  eighths  of  61  ? 

77.  What  is  4  tenths  of  73  ? 

78.  What  is  8  ninths  of  65  ? 

79.  What  is  9  tenths  of  78  ? 

80.  What  is  7  eighths  of  70  ? 

B.     1.  If  you  divide  7  apples  equally  between 

2  boys,  how  many  would  you  give  them  apiece  ? 

2.  How  can  you  divide  5  oranges  equally  among 

3  persons  ? 

3.  Divide  3  bushels  of  corn  equally  among  5  men, 
how  much  would  you  give  them  apiece  ? 

4.  A  boy  had  7  pears,  and  gave  away  3  fifths  ol 
them  ;  how  many  did  he  give  away  ?  and  how  many 
had  he  left  ? 

5.  If  2  yards  of  cloth  cost  3  dollars,  what  is  that 
a  yard  ? 

6.  If  4  yards  of  cloth  cost  2  dollars,  what  is  that 
a  yard  ? 


SECT.  11.]  ARITHMETIC.  99 

[Let  the  answers  be  given  in  dollars  and  cents,  or 
in  shillings.     1  dollar  is  100  cents.] 

7.  If  5  bushels  of  corn  cost  7  dollars,  what  is  that 
a  bushel  ? 

8.  If  a  man  receive  8  dollars  for  6  days'  work, 
what  is  that  per  day  ? 

9.  If  3  bushels  of  wheat  cost  8  dollars,  what  will 

2  bushels  cost  ? 

10.  A   man    had  39  dollars,   and   gave   away  3 
sevenths  of  it ;  how  much  did  he  give  away  ? 

11.  If  4  yards  of  cloth  cost  10  dollars,  what  will 

3  yards  cost  ? 

12.  If  3  barrels  of  cider  cost  8  dollars,  what  will 
10  barrels  cost  ? 

13.  If  7  pounds  of  flour  cost  40  cents,  what  will 
10  pounds  cost  ? 

14.  If  4  firkins  of  butter  cost  26  dollars,  what 
will  7  firkins  cost  ? 

15.  If  3  men  can  do  a  piece  of  work  in  7  days, 
how  long  will  it  take  1  man  to  do  it?    How  long 
would  it  take  4  men  ? 

16.  If  2  cocks  will  empty  a  cistern  in  3  hours,  in 
how  long  a  time  would  1  empty  it  1    In  how  long  a 
time  would  7  cocks  empty  it  ? 


SECTION  XI. 

A.   1.  BOUGHT  1  half  of  a  yard  of  cloth  for  1 
dollar  and  1  half;  what  was  that  a  yard? 

2.  1  and  1  half  is  the  half  of  what  number  ?* 

3.  If  1  third  of  a  yard  of  cloth  cost  1  dollar  and  3 
fourths  of  a  dollar,  how  much  does  a  yard  cost  ? 

*  See  Sect.  VI.  Art.  A.,  and  Sect.  IX  Art.  E^ 


100  ARITHMETIC.  [PART  1. 

4.  2  and  3  fourths  is  I  third  of  what  number  ? 

5.  If  1  half  of  a  barrel  of  beer  cost  2  dollars 
and  1  fourth  of  a  dollar,  how  much  will  a  barrel 
cost? 

6.  2  and  3  fourths  is  one  half  of  what  number  ? 

7.  If  1  fourth  of  a  box  of  lemons  cost  3  dollars 
and  2  thirds  of  a  dollar,  what  will  a  box  cost? 

8.  2  and  2  thirds  is  1  fourth  of  what  number? 

9.  3  and  1  half  is  1  fourth  of  what  number  ? 

10.  If  1  third  of  a  barrel  of  pork  cost  4  dollars 
and  1  third  of  a  dollar,  what  will  1  barrel  cost? 

11.  4  and  2  thirds  is  1  third  of  what  number? 

12.  If  1  sixth  of  a  barrel  of  fish  cost  2  dollars  and 
1  fifth  of  a  dollar,  what  will  a  barrel  cost? 

13.  3  and  2  fifths  is  1  sixth  of  what  number? 

14.  If*  1  fifth  of  a  barrel  of  salmon  cost  3  dollars 
and  2  sevenths  of  a  dollar,  what  is  that  a  barrel  ? 

15.  4  and  3  sevenths  is  1  fifth  of  what  number  ? 

16.  If  a  man  can  travel  4  miles  and  3  sevenths  of 
a  mile  in  1  ninth  of  a  day,  how  far  will  he  travel  in 
a  whole  day  ? 

17.  5  and  3  sevenths  is  1  ninth  of  what  number? 

18.  2  and  4  fifths  is  1  seventh  of  what  number  ? 

19.  6  and  3  eighths  is  1  eighth  of  what  number  ? 

20.  7  and  5  sixths  is  1  third  of  what  number? 

21.  8  and  3  sevenths  is  1  fourth  of  what  number? 

22.  5  and  3  tenths  is  1  seventh  of  what  number  ? 

23.  8  and  5  ninths  is  1  fifth  of  what  number  ? 

24.  9  and  4  fifths  is  1  eighth  of  what  number  ? 

25.  6  and  4  tenths  is  1  tenth  of  what  number  ? 

26.  7  and  4  ninths  is  1  fifth  of  what  number? 

27.  8  and  7  tenths  is  1  seventh  of  what  number  ? 

B.   1.  If  2  thirds  of  a  barrel  of  beef  cost  3  dollars, 
what  does  1  third  of  a  barrel  cost  ? 
2.  3  is  2  times  what  number  ?  * 

*  See  Sect.  VI.,  Art.  B. 


SECT.  11.]  ARITHMETIC.  101 

Ans.  3  is  2  times  the  half  of  3;  but  1  half  of  3  is 
3  halves,  or  1  and  1  half;*  therefore  3  is  2  times  1 
and  1  half. 

3.  If  3  fifths  of  a  yard  of  cloth  cost  4  dollars,  what 
will  1  fifth  of  a  yard  cost  ? 

4.  5  is  3  times  what  number  ? 

5.  If  3  sevenths  of  a  barrel  of  pork  cost  2  dollars, 
what  will  1  seventh  ,of  a  barrel  cost  ? 

6.  2  is  3  times  what  number  ? 

7.  If  4  thirds  of  a  bunch  of  shingles  cost  5  dol- 
lars, what  does  1  third  of  a  bunch  cost  ? 

8.  7  is  four  times  what  number  ? 

9.  2  is  4  times  what  number  ? 

10.  A  man  bought  4  ninths  of  a  barrel  of  flour  for 
three  dollars;  what  would  be  the  price  of  1  ninth 
of  a  barrel  at  the  same  rate  ? 

11.  5  is  4  times  what  number? 

12.  A  man  bought  5  eighths  of  a  hundred  weight 
of  sugar   for  6  dollars ;  what  would  1  eighth  of  a 
hundred  weight  cost  at  the  same  rate  ? 

13.  7  is  5  times  what  number  ? 

14.  3  is  5  times  what  number? 

15.  8  is  5  times  what  number  ? 

16.  9  is  4  times  what  number  ? 

17.  11  is  6  times  what  number  ? 

18.  13  is  7  times  what  number? 

19.  14  is  8  times  what  number? 
29.  17  is  5  times  what  number  ? 

21.  18  is  8  times  what  number? 

22.  17  is  9  times  what  number? 

23.  15  is  10  times  what  number? 

24.  20  is  9  times  what  number  ? 

25.  22  is  10  times  what  number  ? 

26.  24  is  7  times  what  number  ? 

27.  If  2  thirds  of  a  barrel  of  beef  cost  3  dollars, 

*  See  Sect.  X.,  Art.  A. 

9* 


102  ARITHMETIC.  [PART  1. 

what  will  1  third  of  a  barrel  cost  ?  What  will  the 
whole  barrel  cost? 

28.  *  If  5  is  2  thirds  of  some  number,  what  is  1 
third  of  the  same  number?  2  and  1  half  is  1  third  of 
what  number  ?  Then  5  is  two  thirds  of  what  number? 

29.  If  3  fourths  of  a  barrel  of  flour  cost  5  dollars, 
what  will  1  fourth  of  a  barrel  cost  ?  What  will  the 
whole  barrel  cost? 

30.  If  8  is  3  fourths  of  some  number,  what  is  1 
fourth  of  the  same  number  ?  2  and  2  thirds  is  1  fourth 
of  what  number  ?  Then  8  is  3  fourths  of  what  number  ? 

31.  A  man  bought  2  sevenths  of  a  barrel  of  oil  for 
5  dollars ;  how  much  will  1  seventh  cost  at  the  same 
rate  ?  how  much  would  a  barrel  cost  ? 

32.  If  9  is  2  sevenths  of  some  number,  what  is  1  sev-^ 
enth  of  the  same  number  ?  4  and  1  half  is  1  seventh  o£' 
what  number  ?  Then  9  is  2  sevenths  of  what  number  ? 

33.  Bought  5  eighths  of  a  chaldron  of  coal  for  7 
dollars ;  what  is  the  price  of  1  eighth,  at  the  same 
rate  ?  What  is  the  price  of  the  whole  chaldron  ? 

34.  If  12  is  5  eighths  of  some  number,  what  is  1 
eighth  of  the  same  number?    2  and  2  fifths  is  1 
eighth  of  what  number?  Then  12  is  5  eighths  of 
what  number  ? 

35.  If  a  man  can  do  7  ninths  of  a  piece  of  work 
in  4  days,  how  long  would  it  take  him  to  do  1  ninth 
of  it  ?  How  long  would  it  take  him  to  do  the  whole  ? 

36.  If  3  is  7  ninths  of  some  number,  what  is  1 
ninth  of  the  same  number  ?  3  sevenths  is  1  ninth  of 
what  number  ?  Then  3  is  7  ninths  of  what  number  ? 

37.  If  3  fifths  of  a  chaldron  of  coal  cost  8  dollars, 
what  is  the  whole  chaldron  worth  ? 

38.  7  is  3  fifths  of  what  number? 

39.  A  man  bought  6  sevenths  of  a  cask  of  raisins 
for  5  dollars  ;  what  was  the  whole  cask  worth  ? 

40.  8  is  6  sevenths  of  what  number  ? 

*  See  Sect.  VI.,  Art.  B.;  Examples  8  and  following. 


SECT.  11.]  ARITHMETIC.  103 

41.  A  man  had  4  sixths  of  a  week's  board  for  3 
dollars ;  how  much  is  that  for  a  whole  week  ? 

42.  3  is  5  sixths  of  what  number  ? 

43.  9  is  5  eighths  of  what  number  ? 

44.  10  is  7  thirds  of  what  number? 

45.  1 1  is  6  fifths  of  what  number  ? 

46.  12  is  7  fifths  of  what  number? 

47.  15  is  2  sevenths  of  what  number  ? 

48.  17  is  3  tenths  of  what  number? 

49.  A  man  bought  5  eighths  of  a  cask  of  wine  for 
19  dollars;  what  would  the  whole  cask  cost? 

50.  19  is  3  eighths  of  what  number? 

51.  21  is  4  fifths  of  what  number  ? 

52.  Bought  5  ninths  of  a  ton  of  logwood  for  23 
dollars  ;  what  would  a  ton  cost  at  that  rate  ? 

J53.  23  is  7  ninths  of  what  number  ? 

54.  21  is  3  tenths  of  what  number  ? 

55.  29  is  4  sevenths  of  what  number  ? 

56.  31  is  5  eighths  of  what  number  ? 

57.  33  is  6  sevenths  of  what  number  ? 

58.  38  is  4  ninths  of  what  number  ? 

59.  A  man  bought  1  barrel   and  1  seventh  of  a 
barrel  of  wine  for  41  dollars;  what  was  that  a  bar- 
rel? 

NOTE.  1  barrel  and  I  seventh  of  a  barrel  is  the 
same  as  8  sevenths  of  a  barrel  If  8  sevenths  of  a 
barrel  cost  41  dollars,  what  docs  a  barrel  cost  1 

60.  35  is  8  sevenths  of  what  number  ? 

61.  A  man  bought  1  ton  and  4  fifths  of  a  ton  (that 
is,  9  fifths  of  a  ton)  of  fustic  for  43  dollars ;  what 
was  that  a  ton  ? 

62.  52  is  9  fifths  of  what  number  ? 

63.  Bought  1  ton  and  2  sevenths  of  a  ton  of  log* 
wood  for  48  dollars ;  what  was  that  a  ton  ? 

64.  67  is  9  sevenths  of  what  number  ? 

65.  53  is  7  tenths  of  what  number  ? 

66.  58  is  9  eighths  of  what  number  ? 


104  ARITHMETIC.  [PART  1. 

67.  61  is  10  ninths  of  what  number  ? 

68.  Bought  2  barrels  and  1  fifth  of  a  barrel  of  gin 
(that  is,  11  fifths  of  a  barrel)  for  65  dollars;  what 
was  that  a  barrel  ? 

69.  65  is  9  fifths  of  what  number  ? 

70.  71  is  8  elevenths  of  what  number? 

C*  1.  A  boy  gave  away  2  apples  and  1  half, 
which  was  1  fourth  of  all  he  had ;  how  many  had 
he? 

2.  A  man  gave  away  3  dollars,  which  was  2  fifths 
of  all  the  money  he  had ;  how  much  had  he  ? 

3.  A  man  sold  a  cow  for  fifteen  dollars,  which 
was  4  fifths  of  what  she  cost  him ;  how  much  did 
he  lose  by  the  bargain  ? 

4.  A  man  sold  a  piece   of  cloth  for  37  dollars, 
which   was   9  eighths  of  what   it  cost   him;    how 
much  did  he  gain  by  the  bargain  ? 

5.  There  is  a  pole    3  fifths  under  water,  and  7 
feet  out  of  the  water ;  how  long  is  the  pole  ? 

6.  A  man  sold  a  piece  of  cloth  for  47  dollars,  by 
which  bargain  he  lost  2  ninths  of  what  the  cloth 
cost  him ;  how  much  did  it  cost  him,  and  how  much 
did  he  lose? 

Miscellaneous  Examples. 

1.  If  a  staff  5  feet  long  cast  a  shadow  4  feet  at  12 
o'clock,  what  is  the  length  of  a  pole  that  casts  a 
shadow  67  feet  at  the  same  time  ? 

2.  If  53  gallons  of  water,  in  1  hour,  run  into  a 
cistern  containing  97  gallons,  and  44  gallons  run  out 
in  an  hour,  in  what  time  will  it  be  filled  ? 

3.  A  man  bought  a  cask  of  wine  containing  75 
gallons  ;  2  sevenths  of  it  leaked  out,  and  he  sold  the 
remainder  for  1  dollar  a  gallon ;    how  much  did  he 
sell  it  for  ? 


SECT.  11.]  ARITHMETIC.  105 

4.  A  cask  of  wine   cost  67  dollars;    what  is  5 
eighths  of  it  worth  ? 

5.  A  man  bought  9  oranges  for  6  cents  and  2  sev- 
enths apiece,  and  sold  them  for  67  cents ;  what  did 
he  gain  by  the  bargain  ? 

6.  A  man  bought  10  yards  of  broadcloth  for  70 
dollars;    how  must  he  sell  it  per  yard  in  order  to 
gain  14  dollars  1 

7.  If,  when  the  days  are  12  hours  long,  a  man 
perform  a  journey  in  3  days,  how  many  hours  is  he 
in  performing  it  1 

8.  If  a  man  perform  a  journey  in  36  hours,  how 
many  days  would  he  be  in  performing  it,  when  the 
days  are  9  hours  long  ? 

9.  If,  when  the  days  are  11  hours  long,  a  man  can 
perform  a  journey  in  5  days,  in  how  many  hours  will 
he  perform  it  1  In  how  many  days  when  the  days  are 
9  hours  long  ? 

10.  What  number  added  to  2  fifths  of  33  will 
make  the  number  17? 

11.  How  many  yards  of  cloth,  that  is  1  quarter  of 
a  yard  wide,  will  line  10  yards  that  is  3  quarters 
wide? 

12.  8  yards  of  cloth,  that  is  1  quarter  wide,  are 
equal  to  how  many  yards  that  is  4  quarters  wide  ? 

13.  How  many  yards  of  cloth,  that  is  3  quarters 
wide,  are  equal  to  7  yards  that  is  5  quarters  wide  ? 

14.  How  many  yards  of  cloth,  that  is  6  quarters 
wide,  are  equal  to  37  that  is  4  quarters  wide  ? 

15.  If  a  piece  of  cloth  5  quarters  wide  be  worth 
37  dollars,  what  is  a  piece  of  the  same  length,  3 
quarters  wide,  worth? 

16.  If  cloth  4  quarters  wide  is  worth  8  dollars  a 
yard,  what  is  1  yard  of  the  same  kind  of  cloth,  that 
is  5  quarters  wide,  worth  ? 


106  ARITHMETIC.  [PART  1. 


SECTION  XII. 

PARTS  of  one  are  called  fractions.  Fractions  may 
be  expressed  by  figures,  as  well  as  whole  numbers. 
It  requires  two  numbers  to  express  a  fraction;  one 
to  show  into  how  many  parts  one  is  divided,  and 
the  other  to  show  how  many  of  those  parts  are 
used.  For  example,  if  we  wish  to  express  one  half, 
(which  means  that  one  is  divided  into  two  equal 
parts,  and  that  one  part  is  used,)  we  must  use  the 
figure  2  to  express  that  one  is  divided  into  two  equal 
parts,  and  the  figure  1  to  show  that  one  part  is 
used.  And  these  must  be  written  in  such  a  man- 
ner that  we  may  always  know  what  each  of  them 
is  intended  to  express. 

One  half  is  usually  written  thus,  % ;  one  number 
above  a  line,  and  the  other  below  it.  The  number 
below  the  line  shows  into  how  many  parts  one  is 
divided,  and  the  number  above  the  line  shows  how 
many  parts  are  used. 

One  third  is  written -J- 

Two  thirds f 

One  fourth £ 

Three  fourths | 

Two  fifths | 

Example.  $  of  an  apple  signifies  that  the  apple  is 
to  be  cut  into  7  equal  parts,  and  that  3  parts  are 
to  be  used. 

Let  us  apply  an  example  to  Plate  II.  |  refers  to 
a  square  divided  into  8  parts,  and  signifies  that  5 
parts  are  to  be  used. 

We  may  observe,  that,  when  one  is  divided  into 
3  parts,  the  parts  are  called  thirds ;  when  one  is 
divided  into  4  parts,  the  parts  are  called  fourths,  &c. ; 
that  is,  the  fraction  takes  its  name  from  the  number 
of  parts  into  which  one  is  divided.  The  number 


SECT.  12.]  ARITHMETIC.  107 

under  the  line  is  called  the  denominator,  because 
it  gives  name  to  the  fraction ;  and  the  number 
above  the  line  is  called  the  numerator,  because  it 
shows  the  number  of  parts  used.  Thus  -£$,  10  is 
the  denominator,  and  3  the  numerator. 

N.  B.  The  pupil  must  be  made  familiar  with 
this  mode  of  expressing  fractions,  and  must  be  able 
to  apply  it  to  any  familiar  objects,  as  apples,  oran- 
ges, &/c.,  and  to  the  table,  before  he  is  allowed  to 
proceed  any  farther.  Particular  care  must  be  taken 
to  make  him  understand  what  the  denominator  sig- 
nifies, and  what  the  numerator,  as  explained  above. 
The  denominator  should  always  be  explained  first. 

The  following  examples  are  a  recapitulation  of 
some  of  the  foregoing  sections,  for  the  purpose  of 
showing  the  application  of  the  above  method  of 
writing  fractions. 


See  Section  VIII.    A. 

A.     1.  In  2  how  many  times  £  ?     Ans.  £.* 

2.  In  3  how  many  times  £  ?     Ans.  f . 

3.  In  2  how  many  times  ^  1     Ans.  f . 

4.  In  4  how  many  times  -J  ? 

5.  In  6  how  many  times  £  1 

6.  In  7  how  many  times  £  ? 

7.  In  8  how  many  times  -£  ? 

8.  In  2£f  how  many  times  ^  ? 

9.  In  3£  how  many  times  £  1 

10.  Reduce  4-1  to  an  improper  fraction.^ 

11.  Reduce  3f  to  an  improper  fraction. 

12.  Reduce  5f  to  an  improper  fraction. 

*  When  the  numerator  is  larger  than  the  denominator,  the  fraction 
js  called  ati  improper  fraction. 

t  2£  is  read  two  and  1  half.     It  is  called  a  mixed  number. 
\  That  is,  to  find  how  many  fifths  there  are  in  four  and  1  iAh. 


108  ARITHMETIC.  [PART  1. 

13.  Reduce  6|  to  an  improper  fraction. 

14.  Reduce  8T3^  to  an  improper  fraction. 

15.  Reduce  9^  to  an  improper  fraction. 

B.   I.  £  are  how  many  times  1  ? 

2.  f  are  how  many  times  1  ? 

3.  \  are  how  many  times  1  ? 

4.  £  are  how  many  times  1  ? 

5.  f  are  how  many  times  1  ? 

6.  JT&  are  how  many  times  1  ? 

7.  Jjfi-  are  how  many  times  1  ? 

8.  2f-  are  how  many  times  1  ? 

9.  z/-  are  how  many  times  1  ? 
10.  £J  are  how  many  times  1  ? 


See  Section  IX. 

A.  1.  How  much  is  3  times 

2.  How  much  is  4  times  ^  ? 

3.  How  much  is  3  times  f  ? 

4.  How  much  is  4  times  ^  ? 

5.  How  much  is  5  times  f  ? 

6.  How  much  is  6  times  f  ? 

7.  How  much  is  8  times 

8.  How  much  is  9  times 

9.  How  much  is  10  times  ^? 
10.  How  much  is  9  times  f  ? 

B.  1.  How  much  is  3  times 

2.  How  much  is  4  times  3|  ? 

3.  How  much  is  5  times  6|^  ? 

4.  How  much  is  6  times  4  j  ? 

5.  How  much  is  7  times  5f  ? 

6.  How  much  is  8  times  6f  ? 

7.  How  much  is  4  times  lO 


SECT.  12.]  ARITHMETIC.  109 

8.  How  much  is  9  times  7 1  ? 

9.  How  much  is  8  times  9f  ? 
10.  How  much  is  10  times  7-|  ? 


See  Sections  V.  &  X. 

1.  What  is  £  of  6  ?  2.  What  is  J  of  6? 

3.  What  is  |  of  8  ?  4.  What  is  |  of  9  ? 

5.  What  is  f  of  9?  6.  What  is  |  of  10? 

7.  What  is  f  of  14  ?  8.  What  is  |  of  5  ? 

9.  What  is  f  of  5  ?  10.  What  is  £  of  7  ? 

11.  What  is  f  of  7  ?  12.  What  is  f  of  35  ? 

13.  What  is  -i  of  17?  14.  What  is  f  of  26? 

15.  What  is  I  of  27  ?  16.  What  is  |  of  37  ? 

17.  What  is  f  of  47  ?  18.  What  is  f  of  42  ? 

19.  What  is  f  of  65  ?  20.  What  is  f  of  75  ? 


See  Sections  VI.  &  XI. 

A.   1.  2  is  £  of  what  number? 

2.  4  is  £  of  what  number  ? 

3.  8  is  £  of  what  number  ? 

4.  Ij-  is  ±  of  what  number  ? 

5.  2f  is  -J-  of  what  number  ? 

6.  4f  is  ^  of  what  number  ? 

7.  6|-  is  -f  of  what  number  ? 

8.  74  is  %  of  what  number  ? 

9.  8|  is  £  of  what  number? 
10.  9T^j-  is  T\j  of  what  number  ? 


B.   1.  4  is  f  of  what  number? 

2.  6  is  f  of  what  number  ? 

3.  8  is  ^  of  what  number  ? 

10 


110  ARITHMETIC.  [PART  1. 

4.  12  is  ^  of  \fhat  number  ? 

5.  15  is  f  of  what  number  ? 

6.  18  is  f-  of  what  number  ? 

7.  20  Is  f  of  what  number  ? 

8.  24  is  f  of  what  number  ? 

9.  28  is  J  of  what  number  ? 

10.  30  is  T&J  of  what  number  ? 

11.  3  is  §  of  what  number  ? 

12.  4  is  £  of  what  number  ? 

13.  5  is  f  of  what  number  ? 

14.  8  is  g  of  what  number  ? 

15.  9  is  ^  of  what  number  ? 

16.  17  is  f  of  what  number  ? 

17.  25  is  |  of  what  number? 

18.  38  is  f  of  what  number  ? 

19.  43  is  |  of  what  number  ? 

20.  54  is  -y*-  of  what  number  ? 

Miscellaneous  Examples. 

1.  A  man  sold  8  yards  of  cloth  for  3f  dollars  a 
yard ;  what  did  it  come  to? 

2.  A  man  sold  a  horse  for  76  dollars,  which  was 
•§•  of  what  it  cost  him ;  how  much  did  it  cost  him  ? 

3.  A  man  sold  £  of  a  gallon  of  wine  for  40  cents ; 
what  was  that  a  gallon  ? 

4.  If  it  will  take  If  yards  of  cloth  to  make  a  coat, 
how  many  yards  will  it  take  to  make  7  coats  ? 

5.  If  1  horse  consume  3^  bushels  of  oats  in  2 
days,  how  much  would  2  horses  consume  in  5  days  ? 

6.  If,  when  the  days  are  9£  hours  long,  a  man  per- 
form a  journey  in  10  days,  in  how  many  days  would 
he  perform  it  when  the  days  are  12  hours  long? 

7.  A  man  sold  8  yards  of  cloth  for  7-|  dollars  a 
yard,  and  received  8  firkins  of  butter  at  6f  dollars 
a  firkin ;  how  much  was  then  due  to  him  ? 

8.  Two  men  are  38  miles  apart,  and  are  travel- 


SECT.  13.1  ARITHMETIC.  Ill 

ling  towards  each  other,  one  at  the  rate  of  3  miles  an 
hour,  the  other  2  miles ;  how  much  do  they  approach 
each  other  in  an  hour  ?  How  much  in  2  hours  ?  In 
how  many  hours  will  they  meet  ?  At  what  distance 
from  each  place  from  which  they  set  out  ? 


SECTION  XIII. 

A.  1.  If  you  give  ±  of  an  orange  to  one  boy,  and 
£  to  another,  how  much  more  do  you  give  the  first 
than  the  second] 

2.  ±  of  an  orange  is  how  many  £  of  an  orange? 

3.  If  you  give  £  of  an  orange  to  one  boy,  and  £ 
to   another,   how   many  %  would   you   give    away  1 
How  many  £  would  you  have  left  ? 

4.  $  and  £  are  how  many  £? 

5.  A  man  gave  to  one  laborer  ^  of  a  bushel  of 
wheat,  and  f  to  another ;  how  many  %  of  a  bushel 
did  he  give  to  both  ?  How  many  bushels  ? 

6.  J-  and  |  are  how  many  £  ?  How  many  times  1  ? 

7.  A  man  gave  £  of  a  barrel  of  flour  to  one  man, 
and  f  of  a  barrel  to  another ;  to  which  did  he  give 
the  most  ? 

8.  £  is  how  many  £? 

9.  A  man  bought  -£  of  a  bushel  of  wheat  at  one 
time,  and  f  of  a  bushel  at  another ;  at  which  time 
did  he  buy  the  most? 

10.  £  is  how  many  ^  ? 

11.  A  man  bought  |  of  a  yard  of  cloth  at  one 
time,   and  £  of  a  yard   at   another ;  at  which  time 
did  he  buy  the  most  ? 

12.  f  are  how  many  £? 

13.  A  man  wished  to  give'£  of  a  bushel  of  wheat 


112  ARITHMETIC.  [PART  1. 

to  one  man,  and  $  of  a  bushel  to  another ;  but  he 
could  not  tell  how  to  divide  it.  Another  man,  stand- 
ing by,  advised  him  to  divide  the  whole  bushel  into 
six  equal  parts  first,  and  then  take  £  of  them  for 
one,  and  -J  of  them  for  the  other.  How  many  parts 
did  he  give  to  each  1  How  many  to  both  ?  How 
many  had  he  left? 

14.  £  is  how  many  £  ?    £  is  how  many  £  ?    £  and 
J  are  how  many  £? 

15.  A   man,  paying  some  money  to  his  laborers, 
gave  each  man  £  of  a  dollar,  and  each  boy  £  of  a 
dollar ;  how  much  more  did  he  give  to  a  man  than 
to  a  boy? 

16.  What  is  the  difference  between  £  and  £? 

17.  If  a  man  can  earn  £  of  a  dollar  in  a  day,  and 
a  boy  £  of  a  dollar,  how  much  does  the  man  earn 
more  than  the  boy? 

18.  What  is  the  difference  between  £  and  £? 

19.  A  boy,  distributing  some  nuts  among  his  com- 
panions, gave  £  of  a  quart  to  one,  and  £  of  a  quart 
to    another;    how  much    more  did   he  give  to  one 
than  to  the  other  ? 

NOTE.   Change  them  to  sixths. 

20.  What  is  the  difference  between  -J  and  £? 

21.  A  man,  having  two  bushels  of  grain  to  dis- 
tribute among  his  laborers,  wished  to  give  £  of  a 
bushel  to  one,  and  f  of  a  bushel  to  another,  and  the 
rest  to  a  third ;  but  was  at  a  loss  to  tell  how  to  divide 
it ;  at  last  he  concluded  to  divide  each  bushel  into 
six  equal   parts,  or   sixths,  and   then   to  distribute 
those  parts.  How  many  sixths  did  he  give  to  each  ? 

22.  §  is  how  many  £? 

23.  A  man  had  a  horse,  and  a  cow,  and  a  sheep. 
The  horse  would  eat  §  of  a  load  of  hay  in  the  win- 
ter, the  cow  J-,  and  the  sheep  ^.     How  many  £  of  a 
load  would  each  eat  ?     How  many  £  would  they  all 
eat  ?  How  many  loads  ? 


SECT.  13.]  ARITHMETIC.  113 

24.  A  boy,  having  a  quart  of  nuts,  wished  to  di- 
vide them  so  as  to  give  one  companion  £,  another 
J.  and  a  third  £  of  them ;  but,  in  order  to  make  a 
proper   division,  he    first   divided    the   whole    into 
eight  equal  parts,  and  then  he  was  able  to  divide 
them   as   he  wished.      How  many  eighths   did   he 
give  to  each!    How  many  eighths  had  he  left  for 
himself? 

25.  £  is  how  many  £?    £  is  how  many  £?    £  and 
£  and  £  are  how  many  £? 

26.  A  man   gave  f  of  a  barrel   of  flour  to  one 
man,  and  f  of  a  barrel  to  another;  to  which  did  he 
give  the  most  ?  How  much  ? 

27.  Which  is  the  larger,  £  or  |?    How  much  the 
larger  ? 

28.  A  boy,  having  a  pound  of  almonds,  said  he 
intended  to  give  £  of  them  to  his  sister,  and  £  to 
his  brother,  and  the  rest  to  his  mamma.     His  mam- 
ma, smiling,  said  she  did  not  think  he  could  divide 
them  so.     O  yes,  I  can,  said  he ;    I  will  first  divide 
them  into  twelve  equal  parts,  and  then  I  can  divide 
them  well   enough.     Pray  how  many   twelfths   did 
he  give  to  each  ? 

29.  4-  is  how  many  y1^  1   £  is  how  many  T^-  ?    £ 
and  £  are  how  many  yL-  ? 

30.  Mr.  Goodman,  having  a  pound  of  raisins,  said 
he  would  give  Sarah  £,  and  Mary  £,  and  James  £ 
of  them ;    and  he  told  Charles  he  should  have  the 
rest,  if  he  could  tell  how  to  divide  them.  Well,  said 
Charles,  I  would  first  divide  the  whole  into  twelve 
equal  parts,  and  then  I  could  take  £,  and  £,  and  £ 
of  them.     How  many  twelfths  would  each  have? 

31.  •£  and  £  and  £  are  how  many  y1^? 

32.  George    bought   a  pine-apple,    and    said   he 
would  give  £  of  it  to  his  papa,  and  f  to  his  mamma, 
and  y3^  to  his  brother  James,  if  he  could  divide  it. 
James  took  it,  and  cut  it  into  twenty  equal  pieces, 

JO* 


114  ARITHMETIC.  [PART  1. 

and  then  distributed  them  as  George  had  desired. 
How  many  twentieths  did  he  give  to  each  ? 

33.  £  is  how  many  -fa?   -J-  is  how  many  2^?    f  is 
how  many  fa  ?  fa  is  how  many  -fal 

34.  £  is  how  many  - 

35.  £  is  how  many 

36.  £  is  how  many 

37.  £  is  how  many 

38.  f  are  how  many  £  ? 

39.  §  are  how  many 

40.  £  is  how  many  T 

41.  £  are  how  many 

42.  £  are  how  many 

43.  |  are  how  many 

44.  ^  are  how  many  - 

45.  |-  are  how  many 

46.  f  are  how  many 

47.  y3^  are  how  many  ^j  ? 

48.  Reduce  £  to  sixths  and  £  to  sixths* 

49.  f  and  §  are  how  many  £? 

50.  Reduce  £  and  £  to  eighths. 

51.  £  and  £  are  how  many  £? 

52.  \  and  |-  are  how  many  J? 

53.  f  and  £  are  how  many  £  ? 

54.  |^  and  \  are  how  many  £? 

55.  |-  and  f  are  how  many  £  ? 

56.  £  and  f  are  how  many  T\j  ? 

67.  £  and  £  and  £  are  how  many  £  ? 

58.  £  and  £  and  -J^  are  how  many  -fa  1 

59.  §  and  ^  are  how  many  -j-1^  ? 

60.  §  and  £  and  £  are  how  many  -fa  ? 
6L  f  and  -fa  and  £  are  how  many  -fa  ? 

€2.  £  and  |  and  f  and  £  and  ^-  are  how  many  fa  t 

63.  £  and  f  are  how  many  TJ¥  ? 

64.  §  and  -J  are  how  many  y1^  ? 

65.  |  and  f  are  how  many  T^  ? 
€6.  f  less  J  are  how  many  £  7 


SECT.  13.]  ARITHMETIC.  115 

67.  f  and  £  less  T5^,  are  how  many  T^  ? 

68.  £  less  f  are  how  many  rJF  ? 

69.  f  less  f-  are  how  many  ^  ? 

70.  |  less  f  are  how  many  ^  ? 

71.  £,  and  £ ,  and  £,  and  T5^,  less  |,  are  how  many 

&< 

72.  £,  and  £,  and  f ,  and  TV,  and  ^,  less  J,  are 

how  many  ^  ? 

73.  |  and  f  are  how  many  -^1 

74.  f  and  ^  are  how  many  -Jg-  ? 

75.  |-  and  f  are  how  many  ^? 

When  the  denominators  in  two  or  more  frac- 
tions are  the  same,  the  fractions  are  said  to  have  a 
common  denominator.  Thus  f  and  |  have  a  com- 
mon denominator.  We  have  seen  that,  when  two 
of  more  fractions  have  a  common  denominator,  they 
may  be  added  and  subtracted  as  well  as  whole 
numbers.  We  add  or  subtract  the  numerators,  and 
write  their  sum  or  difference  over  the  common 
denominator.  The  first  part  of  the  process  in  the 
above  examples  was  to  reduce  them  to  a  common 
denominator. 

76.  Reduce  f  and  f  to  a  common  denominator. 
NOTE.  They  may  be  reduced  to  twelfths. 

If  it  cannot  be  immediately  seen  what  number 
must  be  the  common  denominator,  it  may  be  found 
by  multiplying  all  the  denominators  together;  for 
that  will  always  produce  a  number  divisible  by  all 
the  denominators. 

77.  Reduce  f  and  -|  to  a  common  denominator. 

78.  Reduce  f  and  J  and  |  to  a  common  denom- 
inator. 

79.  Reduce  ^  and  ^  to  a  common  denominator. 
8Q.  Reduce  ^  and  f  to  a  common  denominator. 


116  ARITHMETIC.  [PART  1. 

81.  Reduce  £  and  f  and  £  to  a  common  denomi- 
nator. 

82.  Add  together  f  and  f . 

83.  Add  together  f  and  ^. 

84.  Add  together  f  and  ^g-. 

85.  Add  together  £  and  £  and  -&. 

86.  Subtract  %  from  £. 

87.  Subtract  -fa  from  f . 

88.  Subtract  |  from  §. 

89.  Subtract  }  from  £. 

B*  1.  Mr.  F.  said  he  would  give  £  of  a  pine-ap- 
ple to  Fanny,  and  f  to  George,  and  the  rest  to  the 
one  that  could  tell  how  to  divide  it,  and  how  much 
there  would  be  left.  But  neither  of  them  could  tell ; 
so  he  kept  it  himself.  Could  you  have  told,  if  you 
had  been  there  ?  How  would  you  divide  it  ?  How 
much  would  be  left  ? 

2.  A  man  sold  l£  bushels  of  wheat  to  one  man, 
4J  bushels  to  another ;   how  many  bushels  did  he 
sell  to  both  ? 

3.  A  man  bought  6J  bushels  of  wheat  at  one 
time,  and  2£  at  another ;  how  much  did  he  buy  in 
the  whole  ? 

4.  A  man  bought  7f  yards  of  one  kind  of  cloth, 
and  6J  yards  of  another  kind ;  how  many  yards  in 
the  whole  ? 

5.  A  man  bought  f  of  a  barrel  of  beer  at  one 
time,  2^  barrels  at  another,  and  6f  at  another ;  how 
much  did  he  buy  in  the  whole  ? 

6.  A  man  bought  one  sheep  for  4§  dollars,  and 
another  for  5^  dollars ;    how  much  did  he  give  for 
both? 

7.  There  is  a  pole  standing,  so  that  ^  of  it  is  in 
the  mud,  and  f  of  it  in  the  water,  and  the  rest  out 
of   the  water;    how  much  of    it   is   out   of   the 
water? 


SECT.  13.]  ARITHMETIC.  117 

8.  A  man,  having  undertaken  to  do  a  piece  of 
work,  did  $  of  it  the  first  day,  -J  of  it  the  second 
day,  and  £  of  it  the  third  day ;  how  much  of  it  did 
he  do  in  three  days  ? 

9.  A  man  having  a  piece  of  work  to  do,  hired 
two  men  and  a  boy  to  do  it.     The  first  man  could 
do  |  of  the  work  in  a  day,  and  the  other  £  of  it, 
and  the  boy  ^  of  it ;  how  much  of  it  would  they  all 
do  in  a  day  ? 

C.  It  will  be  seen,  by  looking  on  Plate  III.  thatf 
is  the  same  as  £,  and  that  f  is  the  same  as  £,  and 
that  |j-  is  the  same  as  f ;  f- ,  f ,  can  therefore  be  re- 
duced to  £,  and  |  to  §.  This  is  called  reducing 
fractions  to  their  lowest  terms. 

1.  Reduce  f  to  its  lowest  terms.*     Ans.  f . 

2.  Reduce  T5^  to  its  lowest  terms. 

3.  Reduce  f  to  its  lowest  terms. 

4.  Reduce  -fa  to  its  lowest  terms. 

5.  Reduce  ^J  to  its  lowest  terms. 

6.  Reduce  T5T  to  its  lowest  terms. 

7.  Reduce  -f^  to  its  lowest  terms. 

8.  Reduce  ^f  to  its  lowest  terms. 

9.  Reduce  -J^-  to  its  lowest  terms. 

10.  Reduce  ^  to  its  lowest  terms. 

11.  Reduce  f  J  to  its  lowest  terms. 

12.  Reduce  §i  to  its  lowest  terms. 

13.  Reduce  £|.  to  its  lowest  terms. 

14.  Reduce  f  J  to  its  lowest  terms. 

Note.  It  will  be  seen  by  the  above  section,  that, 
if  both  the  numerator  and  denominator  be  multi- 
plied by  the  same  number,  the  value  of  the  fraction 
will  not  be  altered ;  or,  if  they  can  both  be  divided 
by  the  same  number  without  a  remainder,  the  frac- 
tion will  not  be  altered. 

*  If  this  article  should  be  found  too  difficult  for  the  pupil,  he  may 
omit  it  till  after  the  next  section. 


118  ARITHMETIC.  [PART  1. 


SECTION  XIV. 

A.  1.  A  BOY,  having  £  of  an  orange,  gave  away  £ 
of  that ;  what  part  of  the  whole  orange  did  he  give 
away? 

2.  What  is  £  of  J? 

3.  If  you  cut  an  apple  into  three  pieces,  and  then 
cut  each  of  those  pieces  into  two  pieces,  how  many 
pieces  will  the  whole  apple  be  cut  into  ?     What  part 
of  the  whole  apple  will  one  of  the  pieces  be  ? 

4.  What  is  £  of  £  ? 

5.  A  boy  had  £  of  a  pine-apple,  and  cut  that  half 
into  three  pieces,  in  order  to  give  away  -J  of  it. 
What  part  of  the  whole  apple  did  he  give  away  ? 

6.  What  is  £  of  £  1 

7.  If  an  orange  be   cut  into  4  parts,  and  then 
each  of  the  parts  be  cut  in  two,  how  many  pieces 
will  the  whole  be  cut  into? 

8.  What  is  £  of  £? 

9.  A  man,  having  £  a  barrel  of  flour,  sold  £  of 
that  •  how  much  did  he  sell  ? 

10.  What  is  ±  of  £? 

11.  If  an  orange  be  cut  into  4  equal  parts,  and 
each  of  those  parts  be  cut  into  3  equal  parts,  how 
many  parts  will  the  whole  orange  be  cut  into? 

12.  What  is  £  of  £  ? 

13.  A  boy,  having  ^  of  a  quart  of  chestnuts, 
gave  away  £  of  what  he  had.     What  part  of  the 
whole  quart  did  he  give  away  ? 

14.  What  is  £  of  £? 

15.  What  is  |  off  1 

16.  A  man,  owning  -J  of  a  ship,  sold  -J  of  his  share ; 
what  part  of  the  ship  did  he  sell,  and  what  part  did 
he  then  own  ? 

17.  What  is  £  of  |? 

18.  What  is  £  of  £? 


SECT.  14.]  ARITHMETIC.  119 

19.  What  is  £  of  |? 

20.  What  is  £  of  £? 

21.  What  is  ^  of  |? 

22.  What  is  £  of  £  ? 

23.  What  is  £  of  |  ? 

24.  What  is  £  of  £  ? 

25.  What  is  |  of  |? 

26.  What  is  -i  of  |? 

27.  What  is  ^  of  £? 

28.  What  is  |  of  £  ? 

29.  A  boy,  having  f  of  an  orange,  (that  is,  2 
pieces,)    gave  his  sister  £  of  what  he  had;    how 
many  thirds  did  he  give  her  ? 

30.  What  is  £  of  §? 

31.  A   boy,  having  f   of   a  pine-apple,  said  he 
would  give  one  half  of  what  he  had  to  his  sister,  if 
she  could  tell  how  to  divide  it.     His  sister  says,  You 
have  got  j-,  or  three  pieces ;  if  you  cut  them  all  in 
two,  you  can  give  me  J-  of  them.     But  £  of  £  is  £ ; 
therefore  I  shall  have  f  of  the  whole  pine-apple. 

32.  What  is  £  of  £? 

33.  A  man,  owning  j-  of  a  share  in  the  Boston 
Bank,  sold  -J-  of  his  part;  what  part  of  a  share  .did 
he  sell  ? 

34.  What  is  £  of  f? 

35.  A  man,  owning   f  of  a  ship,  sold  £  of  his 
share;  what   part  of  the  whole  ship  did   he   sell! 
What  part  had  he  left? 

36.  What  is  £  of  £  ? 

37.  What  is  £  of  |? 

38.  What  is  j  off? 

39.  What  is  |  of  f  ? 

40.  What  is  |  of  J^? 

41.  What  is£of  f  ? 

42.  A  man,  owning  f  of  a  share  in  the  Boston 
Bank,  sold   £  of  his  part;    what  part  of  a  whole 
share  did  he  sell  ? 


120  ARITHMETIC.  [PART  1. 

43.  What  is  £  of  £  ? 

44.  What  is  J  of  f  ? 

45.  A  boy,  having  %  of  a  water-melon,  wished  to 
divide  his  part  equally  between  his  sister,  his  bro- 
ther, and  himself,  but  was  at  a  loss  to  know  how  to 
do  it ;  but  his  sister  advised  him  to  cut  each  of  the 
fifths    into  3  equal   parts.     How  many  pieces    did 
each  have  ?  and  what  part  of  the  whole  melon  was 
each  piece  ? 

46.  What  is  £  of  £  ? 

47.  What  is  £  of  f? 

48.  What  is  |  of  |  ? 

49.  What  is  |  of  |  ? 

50.  What  is  TV  of  |? 

51.  What  is  4  of  J? 

52.  What  is  f  of  £? 

53.  What  is  |  of  |? 

54.  What  is  J  of  f? 

55.  What  is  £  of  f  ? 

56.  What  is  §  of  |  ? 

57.  What  is  |  of  |  ? 

58.  What  is  f  of  f? 

59.  What  is  £  of  f? 

60.  What  is  |  of  f? 

61.  What  is}  of  f  ? 

62.  What  is  $  of  £  ? 

63.  What  is  |  of  f  1 

64.  What  is  T^  of  f  ? 

65.  Whatis-^of  f? 

66.  What  is  f  of -3%  ? 

67.  What  is  I  of  f  ? 

68.  What  is  f  of  f  ? 

69.  If  a  yard  of  cloth  cost  2£  dollars,  what  will 
£  of  a  yard  cost  ? 

'  70.  What  is  £  of  2J? 

71.  A  boy  had  2£  oranges,  and  wished  to  give  £ 
of  them  to  his  sister,  and  £  to  his  brother ;  but  he 


SECT.  14.]  ARITHMETIC.  121 

did  not  know  how  to  divide  them  equally.  His  bro- 
ther told  him  to  cut  the  whole  into  halves,  and 
then  cut  each  of  the  halves  into  3  pieces.  What 
part  of  a  whole  orange  did  each  have  ? 

72.  What  is  |  of  2i? 

73.  A  man  bought  4  bushels  of  corn  for  3§  dol- 
lars; what  part  of  a  dollar  did  1  bushel  cost? 

Change  the  3f  to  thirds,  and  then  find  ^  of  ^  as 
above. 

74.  What  is  £  of  5f? 

75.  If  5  bushels  of  wheat  cost  7f  dollars,  what 
is  that  a  bushel  ? 

76.  What  is  £  of  7f? 

77.  A  man  bought  6  gallons  of  brandy  for  8f  dol- 
lars ;  what  was  that  a  gallon  ? 

78.  What  is  £  of  8£  ? 

79.  A  man  bought  7  gallons  of  wine  for  8f  dol- 
lars ;  how  much  was  that  a  gallon  ? 

80.  What  is  |  of  8f? 

81.  A   man  bought  10  pieces  of  nankin  for  6| 
.^dollars;  how  much  was  it  a  piece? 

82.  What  is  TV  of  6f? 

83.  If  9  bushels  of  rye  cost  7f  dollars,  what  is 
that  a  bushel  ? 

84.  What  is  j.  of  7§  1 

85.  What  is  i  of  5f  ? 

86.  What  is  £  of  8|? 

87.  What  is  £  of  6^  ? 

88.  What  is  |  of  9f  ? 

89.  A  man  bought  7  yards  of  cloth  for  18f  dol- 
lars;   what  was  that  a  yard?    What  would  3  yards 
cost  at  that  rate  ? 

90.  What  is  |  of  18f  ?     What  is  f  of  18|  ? 

91.  A  man  bought  5  barrels  of  cider  for  27f  dol- 
lars ;  what  was  it  a  barrel  ?     What  would  7  barrels 
cost  at  that  rate  ? 

92.  What  is  i  of  27f  ?     What  is  £  of  27f  1 


122  ARITHMETIC.  [PART  1. 

93.  If  6  barrels  of  flour  cost  38f  dollars,  what 
would  10  barrels  cost  at  that  rate  ? 

94.  What  is  Y-of  38£? 

B.  1.  A  man  bought  a  piece  of  cloth  for  42f  dol- 
lars, and  was  obliged  to  sell  it  for  £  of  what  it  cost 
him  ;  how  much  did  he  lose  ? 

2.  A  man  bought  a  quantity  of  flour  for  53^  dol- 
lars,  and  sold  it  for  f  of  what  it  cost  him ;  how 
much  did  he  gain  ? 

3.  If  7  men  can  do  a  piece  of  work  in  4f  days, 
how  long  will  it  take  1  man  to  do  it?    How  long 
will  it  take  3  men  to  do  it? 

4.  If  4  men  can  do  a  piece  of  work  in  9f  days, 
how  long  would  it  take  to  do  it  if  7  men  were  em- 
ployed ? 

5.  There  is  a  pole  standing  so  that  ^  of  it  is  in 
the  water,  and  £  as  much  in  the  mud ;  how  much 
is  in  the  mud  ? 

6.  If  a  man  can  travel  13$-  miles  in  3  hours,  how 
many  miles  will  he  travel  in  8  hours? 

7.  If  5  horses  will  eat  26|-  loads  of  hay  in  a  year, 
what  will  8  horses  eat  in  the  same  time? 

8.  If  4  cocks  will  empty  a  cistern  in  6£  hours, 
how  long  will  it  take  7  cocks  of  the  same  size  to 
empty  it  ? 


SECTION  XV. 

A.  1.  A  BOY,  having  2  oranges,  wished  to  give  £  of 
an  orange  apiece  to  his  playmates ;  how  many  could 
he  give  them  to  ?  If  he  had  given  §  of  an  orange 
apiece,  how  many  could  he  have  given  them  to  ? 

2.  How  many  times  ^  are  there  in  2  ?  How  many 
times  f  are  there  in  2  ? 


SECT.  15.]  ARITHMETIC.  193 

3.  A  man,  having  3  bushels  of  corn,  distributed  it 
among  some  poor  persons,  giving  them  £  of  a  bush- 
el each;  to  how  many  did  he  give  it  ? 

NOTE.   Find  first  how  many  he  would  have  given  it 
to,  if  he  had  given  %  of  a  bushel  to  each. 

4.  In  3  are  how  many  times  £  1  How  many  times  £  ? 

5.  If  f  of  a  barrel  of  flour  will  last  a  family  one 
month,  how  long  will  4  barrels  last  the  same  family  ? 
How  long  will  6  barrels  last?     How  long  will  10 
barrels  last? 

6.  How  many  times  is  f-  contained  in  4?    How 
many  times  in  6  ?  How  many  times  in  10  ? 

7.  If  £  of  a  bushel  of  wheat  will  last  a  family 
one  week,  how  many  weeks  will  6|  bushels  last  the 
same  family  ? 

8.  How  many  times  is  j-  contained  in  6£? 

9.  There  is  a  cistern  having  a  cock  which  will 
fill  it  in  -|  of  an  hour ;    how  many  times  would  the 
cock  fill  the  cistern  in  3f  hours  ? 

10.  How  many  times  is  f  contained  in  3f  ? 

11.  How  much  cloth,  at  l£  dollars  (that  is,  f  dol- 
lars) a  yard,  can  be  bought  for  4  dollars  ? 

12.  How  many  times  is  1^-  or  f  contained  in  4  ? 

13.  A  man  distributed  8£  bushels  of  wheat  among 
some  poor  persons,  giving  1  ±  bushels  to  each ;  how 
many  did  he  give  it  to  1 

14.  How  many  times  is  1  £  contained  in  8^-  ? 

15.  If  a  soldier  is  allowed  1£  pounds  (that"  is,  §  of 
a  pound)  of  meat  in  a  day,  to  how  many  soldiers 
would  6f  pounds  be  allowed  ? 

16.  How  many  times  is  1-J-  contained  in  6f  ? 

17.  If  If  tons  of  hay  will  keep  a  horse  through 
the  winter,  how  many  horses  will  10  tons  keep  ? 

18.  How  many  times  is  If  contained  in  10  ? 

19.  At  2£  dollars  a  box,  how  many  boxes  of  rai- 
sins can  be  bought  for  10  dollars? 

20.  How  many  times  is  2£  contained  in  10? 


134  ARITHMETIC.  [PART  1. 

21.  At  1$  dollars  a  pound,  how  many  pounds  of 
indigo  can  be  bought  for  9f  dollars? 

22.  How  many  times  is  If  contained  in  9f  ? 

23.  At  If  dollars  a  barrel,  how  many  barrels  of 
raisins  can  be  bought  for  9^  dollars  ? 

24.  How  many  times  is  If  contained  in  9£? 

25.  At  J  of  a  dollar  a  piece,  how  many  pieces  of 
nankin  can  be  bought  for  8  J  dollars  ? 

26.  How  many  times  is  J  contained  in  8$  ? 

27.  At  £  of  a  dollar  a  pound,  how  many  pounds 
of  tea  can  be  bought  for  7f  dollars  ? 

28.  How  many  times  is  %  contained  in  7f  ? 

29.  How  many  times  is  3£  contained  in  7f  ? 

30.  How  many  times  is  5£  contained  in  17  ? 

31.  How  many  times  is  4£  contained  in  9f  ? 

32.  How  many  times  is  3^  contained  in  12f  ? 

11.   1.  At^of  a  dollar  a  pound,  how  many  pounds 
of  meat  can  be  bought  for  £  of  a  dollar  ? 
NOTE.     Change  £  to  tenths. 

2.  How  many  times  is  -fa  contained  in  £? 

3.  A  man,  having  £  of  a  barrel  of  flour,  distributed 
it  among  some  poor  persons,  giving  them  £  of   a 
barrel  apiece ;  how  many  did  he  give  it  to? 

NOTE.  Change  both  fractions  to  twelfths;  that  is, 
reduce  them  to  a  common  denominator. 

4.  How  many  times  is  £  contained  in  f  ? 

5.  If  a  pound  of  almonds  cost  ^  of  a  dollar,  how 
many  pounds  can  be  bought  for  f  of  a  dollar? 

NOTE.     Reduce  the  fractions  to  a  common  denomi- 
nator. 

6.  How  many  times  is  ^  contained  in  §  ? 

7.  If  a  piece  of  nankin  cost  f  of  a  dollar,  how 
many  pieces  can  be  bought  for  4f  dollars ;    that  is, 
Y-  dollars  ? 

8.  How  many  times  is  §  contained  in  4  j^  ? 

9.  If  a  bushel  of  barley  cost  f  of  a  dollar,  how 


PART  1.]  ARITHMETIC.  125 

many  bushels  can  be  bought  for  £  of  a  dollar  ?   How 
many  for  If  dollar? 

10.  How  many  times  is  &  contained  in  £  ?     How 
many  times  in  If  1 

11.  How  many  times  is  £  contained  in  f  ? 

12.  How  many  times  is  f-  contained  in  ^1 


TABLES  OF  COINS,  WEIGHTS,  AND  MEASURES. 
Table  I.  —  FEDERAL  MONEY. 

10  mills  make  1  cent. 

10  cents  "  1  dime. 

10  dimes  "  1  dollar. 

10  dollars  "  1  eagle. 

Table  II.  —  STERLING  MONEY. 


4  farthings,  q. 
12  pence 
20  shillings 
6  shillings 
28  shillings 

make 

« 

t( 
u 
ft 

1  penny. 
1  shilling. 
1  pound. 
1  dollar. 
1  guinea. 

s. 
£. 

Table  III.—  TROY  WEIGHT. 

24  grains,  gr.         make        1  pennyweight  dwt. 

20  pennyweights       "            1  ounce.  oz. 

12  ounces                  "            1  pound.  Ib. 

Table  IV.  —  AVOIRDUPOIS  WEIGHT. 

16  drams,  dr.         make         1  ounce.  oz. 

16  ounces                  "            1  pound.  Ib, 

S8  pounds  qr. 


126 


ARITHMETIC. 


[PART  1, 


4  quarters  make     1 

20  hundred  weight     "         1 


hundred  weight, 
ton. 


cwt. 
T. 


2£  inches 
4  nails 

4  quarters 
3  quarters 

5  quarters 

6  quarters 


Table  V. — CLOTH  MEASURE. 
make 


1  nail.  nl. 

1  quarter  of  a  yard.       qr. 

1  yard.  y. 

1  ell  Flemish. 

1  ell  English. 

1  aune,  or  ell  French. 


4  gills 

2  pints 

4  quarts 
3l£  gallons 
63  gallons 

2  hogsheads 

2  pipes 


Table   FJ.— WINE  MEASURE. 


make 


1  pint. 
1  quart. 
1  gallon. 
1  barrel. 
1  hogshead. 
1  pipe. 
1  tun. 


Table   VII. — DRY  MEASURE. 


2  pints 
8  quarts 
4  pecks 


make 


1  quart. 
1  peck. 
1  bushel. 


Table  F///.— MEASURE  OF  TIME. 

60  seconds,  sec.  make  1  minute. 

60  minutes  "  1  hour. 

24  hours  "  1  day. 

7  days  "  1  week. 

4  weeks  "  1  month. 

13  months  1  day  and  \ 

6  hours,  or  365  V    "  1  year, 
days  and  6  hours .; 


pt. 

qt. 

gal. 

bar. 

hd. 


qt. 
pk. 

bu. 


mm. 
h. 
d. 
w. 
m. 


PART  1.]  ARITHMETIC.  127 

For  convenience  of  reckoning,  it  is  usual  in  calen- 
dars to  call  the  year  365  days  for  3  successive  years, 
and  every  fourth  year  366,  (for  in  4  years  the  six 
hours'  overplus  amount  to  a  day,)  which  is  called 
bissextile  or  leap  year.  This  day  is  added  to  Feb- 
ruary. 

The  common  year  is  divided  into  twelve  months, 
which  are  sometimes  called  calendar  months,  be- 
cause they  are  the  months  used  in  calendars. 

The  names  of  the  months,  and  the  number  of 
days  in  each,  are  as  follows  : — 

Names.  Number  of  days. 

I  (    1.  January 31 

•J  (    2.  February 28,  in  leap  year  29. 

ts.  (    3.  March 31 

f  {    4.  April 30 

m  (    5.  May 31 

6.  June 30 

7.  July 31 

8.  August 31 

9.  September    .....  30 
10.  October 31 

•<  I  11.  November  ,  30 


If  12. 

M 


December     .  .  .  .  .31 


Miscellaneous  Examples. 


1 .  In  2  pounds  how  many  ounces  ? 

2.  In  8  yards  how  many  quarters  ? 

3.  In  3  quarters  of  a  yard  how  many  nails? 

4.  t3g.  of  a  dollar  is  how  many  cents  ? 

5.  How  many  farthings  is  |  of  a  penny  ? 

6.  How  many  pence  is  f  of  a  shilling  ? 

7.  f  of  a  yard  is  how  many  quarters  and  nails? 


138  ARITHMETIC.  [PART  1. 

8.  In  £  «£.  how  many  shillings? 

9.  How  much  is  £•  of  a  shilling  ? 

10.  How  much  is  f  of  a  bushel  of  wheat? 

11.  How  much  would  $  of  a  barrel  of  wine  cost, 
at  one  dollar  a  gallon  ? 

12.  How  much  would  £  cwt.  of  sugar  cost,  at  8 
cents  a  pound  ? 

13.  How  much  is  f  of  a  day  ? 

14.  How  much  is  £  of  a  day  ? 

15.  How  much  is  f  of  a  week  ? 

16.  How  much  is  f  of  an  hour  ? 

17.  How  much  would  $  of  a  hogshead  of  wine 
cost,  at  2  dollars  a  gallon  ? 

18.  If  a  man  spend  28  dollars  in  a  month,  what 
is  that  a  week  ?   How  much  a  day  ? 

19.  If  a  man  spend  16  dollars  a  week,  what  is 
that  a  day  ? 

20.  If  a  man  buy  4  bushels  of  grain  for  5  dollars, 
how  much  is  that  a  bushel  ? 

21.  If  wine  is  2  dollars  a  gallon,  how  much  is 
that  a  pint  ? 

22.  If  you  give  5  cents  a  gill  for  wine,  what  is 
that  a  pint?     What  is  it  a  quart?    What  is  it  a  gal- 
lon ? 

23.  If  wine  is  worth  20  cents  a  pint,  what  is  that 
a  gill  ?   What  is  it  a  quart  ?  What  is  it  a  gallon  ? 

24.  If  a  yard  of  cloth  is  worth  7  dollars,  what 
are  2f  yards  worth  ? 

25.  If  a  man  earn  11  dollars  a  week,   what   is 
that  a  day  ?  What  for  3  days  ?  What  for  4J  days  ? 

26.  If  a  man  earn  2f  dollars  in  a  day,  what  will 
he  earn  in  a  week? 

27.  What  is  $  of  a  hogshead  of  wine  ? 

28.  1  farthing  is  what  part  of  a  penny  ? 

29.  2  farthings  is  what  part  of  a  penny  ? 

30.  3  farthings  is  what  part  of  a  penny  ? 
81.  1  penny  is  what  part  of  a  shilling  ? 


PART  1.]  ARITHMETIC.  129 

32.  2  pence  is  what  part  of  a  shilling  ? 

33.  3  pence  is  what  part  of  a  shilling? 

34.  5  pence  is  what  part  of  a  shilling  ? 

35.  6  pence  is  what  part  of  a  shilling  ? 

36.  7  pence  is  what  part  of  a  shilling? 

37.  8  pence  is  what  part  of  a  shilling  ? 

38.  9  pence  is  what  part  of  a  shilling  ? 

39.  10  pence  is  what  part  of  a  shilling  ? 

40.  11  pence  is  what  part  of  a  shilling? 

41.  1  shilling  is  what  part  of  a  pound  ? 

42.  2  shillings  is  what  part  of  a  pound  ? 

43.  3  shillings  is  what  part  of  a  pound  ? 

44.  4  shillings  is  what  part  of  a  pound  1 

45.  5  shillings  is  what  part  of  a  pound  ? 

46.  What  part  of  a  pound  is  6  shillings?    7  shil- 
lings?   8  shillings?    9  shillings  ?    10  shillings?    11 
shillings  ?  12  shillings  ?    13  shillings  ?    14  shillings  ? 
15  shillings  ?    16  shillings  ?    17  shillings  ?    18  shil- 
lings ?  19  shillings  ? 

47.  How  many  farthings  are  there  in  a  shilling? 

48.  One  farthing  is  what  part  of  a  shilling  ? 

49.  2  farthings  is  what  part  of  a  shilling?    3  far- 
things?   4  farthings?    5  farthings?    6  farthings  ?    7 
farthings  ?  8  farthings  ?  9  farthings  ?   10  farthings  ? 

50.  How  many  pence  are  there  in  a  pound  ? 

51.  One  penny  is  what  part  of  a  pound  ? 

52.  What  part  of  a  pound  is  two  pence  ?  3  pence  ? 
4  pence  ?  5  pence  ?  6  pence  ?    7  pence  ?    8  pence  ? 
11  pence  ?  15  pence  ?  27  pence  ?  35  pence? 

53.  How  many  pence  are  there  in  1  shilling  and 
6  pence? 

54.  In  2  shillings  and  4  pence,  how  many  pence  ? 

55.  In  4  shillings  and  5  pence,  how  many  pence  ? 

56.  In  5  shillings  and  8  pence,  how  many  pence  ? 

57.  In  9  shillings  and  11  pence,  how  many  pence  ? 

58.  What  part  of  l£  is  2s.  6d.  ? 

59.  3s.  5d.  is  what  part  of  1^? 


130  ARITHMETIC.  [PART  1. 

NOTE.     Reduce  the  whole  to  pence. 

60.  7s.  8d.  is  what  part  of  1  £1 

61.  What  is  the  price  of  2  yards  of  cloth,  at  3s. 
4d.  a  yard  ? 

62.  What  will  8  yards  of  cloth  cost,  at  2s.  8d.  a 
yard? 

63.  What  will  4  bushels  of  wheat  cost,  at  5s.  9d. 
a  bushel  ? 

64.  What  must  you  give  for  4  barrels  of  cider,  at 
2£  dollars  a  barrel  ? 

65.  If  3  bushels  of  wheat  be  divided  between  two 
men,  how  much  would  they  have  apiece  ? 

66.  If  4  bushels  of  corn  be  divided  among  5  men, 
how  much  would  they  have  apiece  ? 

67.  If  3  bushels  of  corn  be  divided  among  7  men, 
how  much  would  they  have  apiece  ? 

68.  How  many  nails  are  there  in  1  yard  ? 

69.  How  many  nails  are  there  in  4  yards  ? 

70.  How  many  nails  are  there  in  5  yards  and  2 
nails  ? 

71.  In  7  yards  and 3  quarters,  how  many  quarters  ? 

72.  In  4  yards,  2  quarters,  and  3  nails,  how  .many 
nails  ? 

73.  1  nail  is  what  part  of  a  quarter  ? 

74.  3  nails  is  what  part  of  a  quarter  ? 

75.  1  nail  is  what  part  of  a  yard  ? 

76.  What  part  of  1  yard  is  3  nails  ?    5  nails  ?    7 
nails  ?  10  nails  1  15  nails  I 

77.  In  8  quarters  of  a  yard  how  many  yards  1 

78.  In  12  quarters  of  a  yard  how  many  yards  I 

79.  In  10  quarters  of  a  yard  how  many  yards  ? 

80.  In  15  quarters  of  a  yard  how  many  yards  ? 

81.  In  12  nails  how  many  quarters  of  a  yard? 

82.  In  16  nails  how  many  quarters  of  a  yard  ? 
How  many  yards? 

83.  In  24  nails  how  many  quarters  of  a  yard  ? 
How  many  yards  ? 


PART  1.]  ARITHMETIC.  131 

84.  In  35  quarters  of  a  yard  how  many  yards  ? 

85.  In  45  nails  how  many  yards  ? 

86.  In  63  nails  how  many  yards? 

87.  At  2  cents  a  nail,  what  would  4  yards  of  cloth 
cost? 

88.  At  2f  dollars  for  1  quarter  of  a  yard,  what 
would  2  yards  cost  ? 

89.  1  oz.  is  what  part  of  a  Ib.  ? 

90.  What  part  of  a  Ib.  is  2  oz.  ?  3  oz.  ?  4  oz.  ?  5 
oz.?  7oz.?  10  oz.?  15  oz.? 

91.  What  part  of  a  qr.  of  1  cwt.  is  1  Ib.  ?  2  Ibs.  ? 
31bs.?    4  Ibs.?    7  Ibs.?    9  Ibs.  ?    J41bs.  ?    18  Ibs.  ? 
23  Ibs.  ? 

92.  At  3  cents  for  1  oz.  what  would  1  Ib.  cost  ? 

93.  At  3  cents  for  2  oz.  what  would  1  Ib.  cost? 

94.  At  3  cents  for  8  oz.  what  would  1  Ib.  cost? 

95.  At  5  cents  for  10  oz.  what  would  1  Ib.  cost? 

96.  At  8  shillings  for  4  Ibs.  what   would   10   Ibs. 
cost? 

97.  If  a  man  consume  1  Ib.  and  3  oz.  of  meat  in 
a  day,  how  much  would  he  consume  in  a  week  ? 

98.  If  a  man  spend  2f   dollars  in  a  day,  how 
much  would  he  spend  in  a  week  ? 

99.  If  a  man  travel  3f  miles  in  an  hour,  how  far 
would  he  travel  in  3  hours  ?    How  far  in  7  hours  ? 
How  far  in  12  hours  ? 

100.  If  2  men  start  from  the  same  place,  and  trav- 
el in  opposite  directions,  one  at  the  rate  of  3f  miles 
in  an  hour,  and  the  other  4-i  miles,  how  far  will 
they  be  apart  at  the  end  of  1  hour  ?    How  far  at  the 
end  of  2  hours  ?    How  far  at  the  end  of  3  hours  ? 
How  far  at  the  end  of  7  hours  ? 

101.  Two   men  start  from  the  same  place,  and 
travel  the  same  way,  one  at  the  rate  of  4-f  miles  in 
an  hour,  the  other  at  the  rate  of  4f  miles  in  an  hour ; 
how  far  will  they  be  apart  at  the  end  of  1  hour  ? 
How  far  in  2  hours  ?  How  far  in  5  hours  ?  How  far  in 


132  ARITHMETIC.  [PART  1. 

10  hours?   How  far   in   3   days,  if  they   travel   10 
hours  in  a  day  ? 

102.  How  many  yards  of  cloth,  at  5  dollars  a 
yard,  must  be  given  for  8  barrels  of  flour,  at  7  dol- 
lars per  barrel  ? 

103.  What  part  of  a  month  is  1  week  ?    2  weeks  ? 
3  weeks  ? 

104.  What  part  of  a  year  is  1  month  1  2  months  ? 
3  months  ?    4  months  1    5  months  ?    6  months  ?    7 
months?   8  months?    9  months?    10  months?    11 
months  ? 

105.  What  part  of  1  month  is  1  day  ?   2  days  T 
3  days?   7  days?   8  days?    11  days?    15  days?    18 
days  ?  20  days?  24  days  ?  37  days  ? 

106.  If  5    bushels  of  oats   will   keep   7   horses 
through  the  winter,  how  many  bushels  will  it  take 
to  keep  12  horses  the  same  time  ? 

107.  If  you  give  7  men  2-J  bushels  of  corn  apiece, 
how  many  bushels  would  it  take  for  the  whole  ? 

108.  A  man,  failing  in  trade,  was  able  to  pay  his: 
creditors  only  4  shillings  on   a  dollar ;  how  much 
would  he  pay  on  2  dollars  ?  How  much  on  3  dollars  ? 
How  much  on  7  dollars?  How  much  on  10  dollars? 

109.  A  man,  failing  in  trade,  is  able  to  pay  only  9 
shillings  on  a  pound ;  how  much  would  he  pay  on 
a  debt  of  2  pounds  ?  How  much  on  3  pounds  ?  How 
much  on  12  pounds  ? 

110.  A  man,  failing  in  trade,  is  able  to  pay  only  4 
shillings  and  7  pence  on  a  dollar ;  how  much  would 
he  pay  on  a  debt  of  7  dollars? 

111.  If  6  dollars'  worth  of  provisions  will  serve  3 
men  5  days,  how  many  days  will  it  serve  1  man  ? 
How  many  days  will  it  serve  2  men?    How  many 
days  will  it  serve  8  men  ? 

112.  If  10  dollars'  worth  of  provision  will  serve  7 
men  4  days,  how  many  days  will  it  serve  9  men? 

113.  If  12  dollars'  worth  of  provisions  will  serve  7 


PART  1.]  ARITHMETIC.  133 

men  3  days,  how  many  men  would  it  serve  1  day? 
How  many  2  days  ?    How  many  8  days? 

114.  If  11  dollars'  worth  of  provision  will  serve  6 
men  8  days,  how  many  men  will  it  serve  5  days  ? 

115.  If  8  dollars'  worth  of  provisions  will  serve  7 
men   5   days,    how   many   days   would   16   dollars' 
worth  of  provision  last  4  men  ? 

116.  If  1   peck   of  wheat  afford  12   ten-penny 
loaves,  how   many  penny  loaves   may  be   obtained 
from  it  ?    How  many  two-penny  loaves  ?    How  many 
three-penny  loaves?  How  many  seven-penny  loaves  ? 

117.  If  1  peck  of  wheat  afford  11  eight-penny 
loaves,  how  many  ten-penny  loaves  will  it  afford? 

118.  A  man,  having  hired  some  men  and  some 
boys,  agreed  to  give  each  man  3  shillings,  and  each 
boy  2  shillings ;    how  much  would  it  take  to  pay  a 
man  and  a  boy  ?    How  much  2  men  and  2  boys  ? 
How  much  7  men  and  7  boys? 

119.  A  man,  having  18  shillings  to  pay  among 
his  laborers,  would  give  to  every  man  2  shillings,  and 
to  every  boy  1    shilling;    the  number  of  men  and 
boys  was  equal ;  how  many  were  there  of  each  ? 

120.  A  gentleman,  having    50   shillings   to    pay 
among   his  laborers,   would   give   to   every  man   8 
pence,  and  to  every  boy  4  pence ;  the  number  of 
men  and  boys  was  equal ;  how  many  were  there  of 
each  ? 

121.  Two  men  bought  a  bushel  of  corn ;  one  gave 
1  shilling,  the  other  2  shillings ;  what  part  of  the 
whole  did  each  pay  ?     What  part  of  the  corn  must 
each  have  ? 

122.  Two  men  bought  a  barrel  of    flour  for  8 
dollars;  one  gave   3   dollars,  the   other   5   dollars; 
what  part  did  each  pay  ?  and  what  part  must  each 
have? 

123.  Three  men,  A,  B,  and  C,  hired  a  garden  ; 
A  paid  6  dollars,  B  5  dollars,  and  C  9  dollars ;  how 

12 


134  ARITHMETIC.  [PART  1. 

much  did  they  all  pay  ?  What  part  of  the  whole 
did  each  pay?  They  sold  the  produce  for  forty 
dollars ;  what  part  of  it  must  each  have  ?  What 
did  each  one's  share  amount  to? 

124.  Three  men   bought  a  lottery  ticket  for  10 
dollars;  the  first  gave  3  dollars;  the  second  5  dol- 
lars, and  the  third  2  dollars.     They  drew  a  prize  of 
120  dollars.     What  was  each  man's  share  ? 

125.  Two  men  hired  a  pasture  for  58  dollars; 
one  put  in  7  horses,  and  the  other  3  horses ;  what 
ought  each  to  pay  ? 

126.  Three  men  commenced  trade  together;  they 
put  in  money  in  the  following  proportion  ;  the  first, 
3  dollars,  as  often  as  the  second  put  in  4,  and  as 
often  as  the  third  put  in  5 ;  they  gained  87  dollars. 
What  was  each  man's  share  of  the  gain  ? 

127.  Two  men  hired  a  pasture  for  32  dollars  ; 
the  first  put  in  3  sheep  for  4   months ;  the  second 
put  in  4  sheep  for  five  months ;  how  much  ought 
each  to  pay  ? 

Note.  Three  sheep  for  four  months  is  the  same 
as  12  sheep  for  1  month ;  4  sheep  for  5  months  is 
the  same  as  20  sheep  for  1  month.  This  question 
is  therefore  the  same  as  if  one  man  put  in  12  sheep, 
and  the  other  20  sheep. 

128.  Two  men,  A   and  B,  traded  in  company  ; 
A  put  in  1  dollar  for  4  months,  and  B  2  dollars  for 
3  months,  and  they  gained  ninety  cents ;  how  many 
cents  must  each  have  ? 

129.  Three  men,  A,  B,  and  C,  traded  in  compa- 
ny, and  put  in  money  in  the  following  proportions  ; 
A  put  in  4  dollars  as  often  as  B  put  in  3,   and  as 
often  as  C  put  in  2 ;  A's  money,  was  in  2  months, 
B's  3  months,  and  C's  4  months,  and  they   gained 
100  dollars  ;  what  was  each  one's  share  ? 

130.  Two  men,  A  and  B,  traded   in  company  ; 
A  put  in  2  dollars  as  often  as  B  put  in  3  ;  A's  money 


PART  1.]  ARITHMETIC.  135 

was  employed  7  months,  and  B's  money  5  months ; 
they  gained  58  dollars;  what  was  each  man's  share 
of  the  gain  ? 

1-M.  Three  men,  A,  B,  and  C,  traded  in  company, 
and  put  in  money  in  the  following  proportions ;  A 
put  in  %  dollars  as  often  as  B  put  in  4,  and  as  often 
as  C  put  in  6 ;  B's  money  was  in  twice  as  long  as 
C's,  and  A's  two  times  as  long  as  B's  ;  they  gained 
88  dollars ;  what  was  each  one's  share  of  the  gain  ? 

NOTE.  Interest  is  a  reward  or  premium  allowed  by 
a  debtor  to  a  creditor  for  the  use  of  money.  The 
usual  interest  for  1  year,  and  that  which  is  establish- 
ed by  law  in  this  country,  is  6  cents  on  a  dollar,  6 
dollars  on  a  hundred  dollars,  or  6  pounds  on  a  hun- 
dred pounds  ;  or,  in  fine,  T^  of  the  sum,  whatever  be 
the  denomination.  It  is  called  6  per  cent.,  that  is,  6 
on  the  hundred,  because  it  is  always  reckoned  by  the 
hundred.  So  3  per  cent.,  4  per  cent.,  fyc.  signify 
Ttf^j  Ttftf'  4*c->  or  so  much  on  a  hundred. 

132.  The  interest  of  1  dollar  being  6  cents  for  1 
year,  what  is  the  interest  of  7  dollars  for  the  same 
time  ?     What  is  the  interest  of  10  dollars  ?     Of  15 
dollars?    Of 20  dollars?    Of  30  dollars?    Of 50  dol- 
lars? Of  75  dollars?  Of  100  dollars?  Of  118 dollars? 

133.  If  the  interest  of  1  dollar  is  6  cents  for  1 
year,  what  would  it  be  for  2  years  ?     What  would  be 
the  interest  of  8  dollars  for  two  years  ?     Of  17  dol- 
lars?    Of  43  dollars? 

134.  If  the  interest  of  100  dollars  is  6  dollars  for 
a  year,  what  would  be  the  interest  of  50  dollars  for 
trie  same  time?   Of  2  hundred?  Of  3  hundred  ?    Of 
4  hundred  ?     Of  1  hundred  and  50  ?    Of  2  hundred 
and  50? 

135.  If  the  interest  of  100  dollars  is  six  dollars 
for  1  year,  what  would  be  the  interest  of  it  for  6 
months?     For  3  months?     For  4  months?     For  8 


136  ARITHMETIC.  [PART  1. 

months?  For  9  months?  For  1  month?  For  2 
months  ?  For  5  months  ?  For  7  months  ?  For  10 
months?  For  11  months? 

136.  What  is  the  interest  of  100  and  32  dollars 
for  2  years,  at  6  per  cent.  ? 

137.  What  is  the  interest  of  300  dollars   for  1 
year  and  6  months,  at  6  per  cent.? 

138.  What  is  the  interest  of  1  dollar  for  6  months, 
at  6  per  cent.  ?     What  for  2  months  1     What  for  1 
month  ?  What  for  3  months  ?  4  months  ?  5  months  ? 
7  months?  9  months?  11  months? 

139.  What  is  the  interest  of  57  dollars  for  1  year 
and  7  months,  at  6  per  cent.  ? 

140.  What  is  the  interest  of  200  and  67  dollars 
for  1  year  and  4  months,  at  6  per  cent.  ? 

141.  If  the  interest  of  1  year  is  6  per  cent.,  what 
would  be  the  per  cent,  for  2  years  ?     For  3  years  ? 
For  6  months  ?    For  2  months  ?    For  1  month  ?    For 
4  months  ?    For  5  months  ?    For  7  months  ?   For  8 
months  ?    For  9  months  ? 

142.  If  the  interest  of  2  months,  or  60  days,  is  1 
per  cent.,  what  would  be  the  per  cent,  for  20  days? 
What  for  40  days?     What  for  15  days?     What  for 
45  days?     What  for  12  days?     What  for  10  days? 
What  for  5  days  ? 

143.  What  is  the  interest  of  100  and  37  dollars 
for  2  years  3  months  and  20  days  ? 

144.  A  can  do  a  piece  of  work  in  2  days ;  how 
much  of  it  can  he  do  in  1  day  ? 

145.  B  can  do  a  piece  of  work  in  4  days ;  how 
much  of  it  can  he  do  in  1  day  ? 

146.  If  A  can  do  ^  of  a  piece  of  work  in  1  day, 
and  B  can  do  £  of  it  in   1   day,  how  much  would 
both  do  in  a  day  ?     How  long  would  it  take  them 
both  together  to  do  the  whole  ? 

147.  If  1  man  can  do  a  piece  of  work  in  2  days, 
and  another  in  3  days,  how  much  of  it  would  each 


PART  1.]  ARITHMETIC.  137 

do  in  a  day?     How  much  would  both  together  do? 
How  long  would  it  take  them  both  to  do  the  whole  ? 

148.  A  cistern  has  2  cocks  ;  the  first  will  fill  it  in 
3  hours,  the  second  in  6  hours ;    how  much  of  it 
would  each  fill  in  an  hour?     How  much  would  both 
together  fill  ?     How  long  would  it  take  them  both  to 
fill  it  ? 

149.  A  man  and  his  wife  found  by  experience, 
that,  when  they  were  both  together,  a  bushel  of  meal 
would  last  them  only  2  weeks ;  but  when  the  man 
was  gone,  it  would  last  his  wife  5  weeks.     How  much 
of  it  did  both  together  consume  in  1  week  ?     What 
part  did   the   woman    alone    consume    in    1   week? 
What  part  did  the  man  alone  consume  in  one  week? 
How  long  would  it  last  the  man  alone  ? 

150.  If  1  man  could  build  a  piece  of  wall  in  5 
days,  and   another  man  could  do  it  in  7  days,  how 
much  of  it  would  each  do  in  1   day  1     How  many 
days  would  it  take  them  both  do  it  ? 

151 .  A  cistern  has  3  cocks  ;  the  first  would  fill  it 
in  3  hours,  the  second  in  6  hours,  the  third  in  4 
hours ;  what  part  of  the  whole  would  each  fill  in  1 
hour  ?  and  how  long  would  it  take  them  all  to  fill  it, 
if  they  were  all  running  at  once? 

15*2.  A  and  B  together  can  build  a  boat  in  8  days, 
and  with  the  assistance  of  C  they  can  do  it  in  5 
days;  how  much  of  it  can  A  and  B  build  in  1  day? 
How  much  of  it  can  A,  B,  and  C,  build  in  1  day  ? 
How  much  of  it  can  C  build  alone  in  1  day  ?  How 
long  would  it  take  C  to  build  it  alone  ? 

153.  Suppose  I  would  line  8  yards  of  broadcloth 
that  is  1£  yards  wide,  with  shalloon  that  is  £  of  a 
yard  wide ;    how  many  yards  of  the  shalloon  will 
line  1  yard  of  the  broadcloth  ?  How  many  yards  will 
line  the  whole  ? 

154.  If  7  yards  of  cloth  cost  13  dollars,  what  will 
10  yards  cost? 

12* 


138  ARITHMETIC.  [PART  1. 

155.  If  the  wages  of  25  weeks  come  to  75  dollars, 
what  will  be  the  wages  of  7  weeks? 

156.  If  8  tons  of  hay  will  keep  7  horses  three 
months,  how  much  will  keep   12  horses  the  same 
time? 

157.  If  a  staff  4  feet  long  cast  a  shadow  6  feet 
long,  what  is  the  length  of  a  pole  that  casts  a  shadow 
58  feet  at  the  same  time  of  day  1 

158.  If  a  stick  8  feet  long  cast  a  shadow  2  feet  in 
length,  what  is  the  height  of  a  tree  which  casts  a 
shadow  42  feet  at  the  same  time  of  day? 

159.  At  6  dollars  per  week,  how  many  months' 
board  can  I  have  for  100  dollars  ? 

160.  A  ship  has  sailed  24  miles  in  4  hours  ;    how 
long  will  it  take  her  to  sail  150  miles  at  the  same 
rate? 

161.  30  men  can  perform  a  piece  of  work  in  20 
days ;  how  many  men  will  it  take  to  perform  the 
same  work  in  8  days  ? 

162.  17  men  can  perform  a  piece  of  work  in  25 
days ;  in  how  many  days  would  5  men  perform  the 
same  work? 

163.  A  hare  has  76  rods  the  start  of  a  greyhound, 
but  the  greyhound  runs  15  rods  to  ten  of  the  hare ; 
how  many  rods  must  the  greyhound  run  to  overtake 
the  hare  ?  % 

164.  A  garrison  has  provision  for  8  months,  at 
the  rate  of  15  ounces  per  day ;  how  much  must  be 
allowed  per  day,  in  order  that  the  provision  may  last 
11  months? 

165.  If  8  men  can  build  a  wall  15  rods  in  length 
in  10  days,  how  many  men  will  it  take  to  build  a 
wall  45  rods  in  length  in  5  days  ? 

166.  If  a  quarter  of  wheat  affords  60  ten-penny 
loaves,  how  many  eight-penny  loaves  may  be  ob- 
tained from  it? 

167.  Said  Harry  to  Dick,  My  purse  and  money 


PART  1.]  ARITHMETIC.  139 

together  are  worth  16  dollars,  but  the  money  is 
worth  7  times  as  much  as  the  purse ;  how  much 
money  was  there  in  the  purse?  and  what  is  the 
value  of  the  purse  ? 

168.  A  man,  being  asked  the  price  of  his  horse, 
answered,  that  his  horse  and  saddle  together  were 
worth  100  dollars,  but  the  horse  was  worth  9  times 
as  much  as  the  saddle.     What  was  each  worth  ? 

169.  A  man,  having  a  horse,  a  cow,  and  a  sheep, 
was  asked  what  was  the  value  of  each.    He  answer- 
ed, that  the  cow  was  worth  twice  as  much  as  the 
sheep,  and  the  horse  3  times  as  much  as  the  sheep, 
and  that  all  together  were  worth  60  dollars.      What 
was  the  value  of  each  1 

170.  A  man  bought  an  apple,  an  orange,  and  a 
melon,  for  21  cents;    for  the  orange  he  gave  twice 
as  much  as  for  the  apple,  and  for  the  melon  he  gave 
twice  as  much  as  for  the  orange.     How  much  did 
he  give  for  each  ? 

171.  If  80  dollars'  worth  of  provision  will  serve 
20  men  24  days,  how  many  days  will   100  dollars' 
worth  of  provisions  serve  30  men  1 

172.  There  is  a  pole  £  and  %  under  water,  and 
10  feet  out ;  how  long  is  the  pole  ? 

173.  In  an  orchard  of  fruit  trees,  £  of  them  bear 
apples,  j-  of  them  bear  plums,  -J-  of  them  pears,  7 
of  them  peaches,  and   3   of  them   cherries;    how 
many  trees  are  there  in  the  whole,  and  how  many 
of  each  sort  ? 

*"  174.  A  farmer,  being  asked  how  many  sheep  he 
had,  answered,  that  he  had  them  in  4  pastures  ;  in 
the  first  he  had  ^  of  his  flock ;  in  the  second  £ ;  in 
the  third  £ ;  and  in  the  fourth  15 ;  how  many  sheep 
had  he  ? 

175.  A  man,  driving  his  geese  to  market,  was 
m.et  by  another,  who  said,  Good  morrow,  master, 
with  your  hundred  geese ;  says  he,  I  have  not  a 


140  ARITHMETIC.  [PART  1. 

hundred ;  but  if  I  had  half  as  many  more  as  I  now 
have,  and  two  geese  and  a  half,  I  should  have  a 
hundred  ;  how  many  had  he  ? 

176.  What  number  is  that,  to  which  if  its  half  be 
added,  the  sum  will  be  60  ? 

177.  What  number  is  that,  to  which   if  its  third 
be  added,  the  sum  will  be  48? 

178.  What  number  is  that,  to  which  if  its  fifth 
be  added,  the  sum  will  be  54  ? 

179.  What  number  is  that,  to  which  if  its  half 
and  its  third  be  added,  the  sum  will  be  55? 

180.  A  man,  being  asked  his  age,  answered,  that, 
if  its  half  and  its  third  were  added  to  it,  the  sum 
would  be  77 ;  what  was  his  age  ? 

181.  What  number  is  that,  which  being  increased 
by  its  half,   its  fourth,  and   eighteen  more,  will   be 
doubled  ? 

182.  A  boy,  being  asked  his  age,  answered,  that, 
if  $  and  £  of  his  age,  and  20  more,  were  added  to 
his  age,  the  sum  would  be  3  times  his  age.     What 
was  his  age  ? 

183.  A  man,  being  asked  how  many  sheep  he  had, 
answered,  that,  if  he  had  as  many  more,  £  as  many 
more,  and  2£  sheep,  he  should  have    100.     How 
many  had  he? 


ARITHMETIC. 


FAKT    II. 


KEY. 

THE  Key  contains  an  explanation  of  the  plates, 
and  the  manner  of  using  them.  The  manner  of 
solving  the  examples  in  each  section  is  particularly 
explained.  All  the  most  difficult  of  the  practical  ex- 
amples are  solved  in  such  a  manner,  as  to  show  the 
principles  by  which  they  are  performed.  Care  has 
been  taken  to  select  examples  for  solution,  that  will 
explain  those  which  are  not  solved.  Many  remarks, 
with  regard  to  the  manner  of  illustrating  the  prin- 
ciples to  the  pupils,  are  inserted  in  their  proper 
places. 

Instructers  who  may  never  have  attended  to  frac- 
tions, need  not  be  afraid  to  undertake  to  teach  this 
book.  The  author  flatters  himself  that  the  princi- 
ples are  so  illustrated,  and  the  processes  are  made  so 
simple,  that  any  one,  who  shall  undertake  to  teach 
it,  will  find  himself  familiar  with  fractions  before  he 
is  aware  of  it,  although  he  knew  nothing  of  them 
before ;  and  that  every  one  will  acquire  a  facility  in 
solving  questions  which  he  never  before  possessed. 

The  reasoning  used  in  performing  these  small 
examples  is  precisely  the  same  as  that  used  upon 
large  ones.  And  when  any  one  finds  a  difficulty  in 
solving  a  question,  he  will  remove  it  much  sooner, 


142  KEY  [PART  2. 

and  much  more  effectually,  by  taking  a  very  small 
example  of  the  same  kind,  and  observing  how  he 
does  it,  than  by  recuVring  to  a  rule. 

The  practical  examples  at  the  commencement  of 
each  section  and  article,  are  generally  such  as  to 
show  the  pupil  what  the  combination  is,  and  how  he 
is  to  perform  it.  This  will  teach  the  pupil  gradual- 
ly to  reason  upon  abstract  numbers.  In  each  com- 
bination, there  are  a  few  abstract  examples  without 
practical  ones,  to  exercise  the  learner  in  the  com- 
binations, after  he  knows  what  these  combinations 
are.  It  would  be  an  excellent  exercise  for  the  pu- 
pil to  put  these  into  a  practical  form  when  he  is  re- 
citing. For  instance,  when  the  question  is,  How 
many  are  5  and  3?  let  him  make  a  question  in  this 
way  :  If  an  orange  cost  5  cents,  and  an  apple  3  cents, 
what  would  they  both  come  to?  This  may  be  done 
in  all  cases. 

The  examples  are  often  so  arranged,  that  several 
depend  on  each  other,  so  that  the  preceding  ex- 
plains the  following  one.  Sometimes,  also,  in  the 
same  example,  there  are  several  questions  asked, 
so  as  to  lead  the  pupil  gradually  from  the  simple  to 
the  more  difficult.  It  would  be  well  for  the  pupil 
to  acquire  the  habit  of  doing  this  for  himself,  when 
difficult  questions  occur. 

The  plates  should  be  used  for  young  pupils,  but 
they  are  not  necessary  for  the  older  ones.  The 
plates  for  fractions,  however,  will  frequently  be  use- 
ful to  these.  The  first  plate  need  not  be  used  much, 
after  the  pupil  is  familiar  with  the  multiplication 
table. 

The  book  may  be  used  in  classes,  where  itds  con- 
venient. The  pupil  may  answer  the  questions  with 
the  book  before  him  "or  not,  as  the  instructor  thinks 
proper.  A  very  useful  mode  of  recitation  is,  for 
the  iristructer  to  read  the  example  to  the  whole 


PART  2.]  KEY.  143 

class,  and  then,  allowing  sufficient  time  for  them  to 
perform  the  question,  call  upon  some  one  to  answer 
it.  In  this  manner  every  pupil  will  be  obliged  to 
perform  the  example,  because  they  do  not  know 
who  is  to  answer  it.  In  this  way  it  will  be  best  for 
them  to  answer  without  the  book. 

It  will  often  be  well  to  let  the  elder  pupils  hear 
the  younger.  This  will  be  a  useful  exercise  for 
them,  and  an  assistance  to  the  instructer. 

Explanation  of  Plate  I. 

This  plate,  viewed  horizontally,  presents  ten  rows 
of  rectangles,  and  in  each  row  ten  rectangles. 

In  the  first  row,  each  rectangle  contains  one  mark, 
each  mark  representing  unity  or  one.  In  the  sec- 
ond row,  each  rectangle  contains  two  marks ;  in  the 
third,  three  marks,  &c. 

The  purpose  of  this  plate  is,  first,  to  represent 
unity  either  as  a  unit,  or  as  making  a  part  of  a  sum 
of  units:  secondly,  to  represent  a  collection  of 
units,  either  as  forming  a  unit  itself,  or  as  making  a 
part  of  another  collection  of  units  ;  and  thus  to  com- 
pare unity  and  each  collection  of  units  with  another 
collection,  in  order  to  ascertain  their  ratios. 

All  the  examples  as  far  as  the  eighth  section  can 
be  solved  by  this  plate.  The  manner  of  using  it  is 
explained  in  the  Key  for  each  section  in  its  proper 
place. 

The  pupil,  if  very  young,  should  first  be  taught 
to  count  the  units,  and  to  name  the  different  assem- 
blages of  units,  in  the  following  manner  : — r 

The  instructer,  showing  him  the  first  row,  which 
contains  ten  units  insulated,  requests  the  pupil  to 
put  his  finger  on  the  first,  and  say,  one  ,*  then  on  the 
second,  and  say,  and  one  are  two ;  and  on  the  third,  and 
say,  and  one  are  three ;  and  so  on.  to  ten  :  then^  com- 


144  KEY.  [PART  2. 

mencing  the  row  again,  let  him  continue  and  say, 
ten  and  one  are  eleven,  &c. 

After  adding  them,  let  him  begin  with  ten,  and 
say,  ten  less  one  are  nine,  nine  less  one  are  eight,  &/c. 
Then,  taking  larger  numbers,  as  twenty  or  thirty, 
let  him  subtract  them  in  the  same  manner. 

Next,  let  him  name  the  different  assemblages,  as 
twos,  threes,  &,c.  Afterwards,  let  him  count  the 
number  of  units  in  each  row. 

Note.  The  sections,  articles,  and  examples,  are 
referred  to  by  the  same  marks  which  distinguish 
them  in  Part  I. 


SECTION  I. 

A.  THIS  section  contains  addition  and  subtraction. 
The  first  examples  may  be  solved  by  means  of  beans, 
peas,  &/c.,  or  by  Plate  I.  The  former  method  is 
preferable,  if  the  pupil  be  very  young,  not  only  for 
the  examples  in  the  first  part  of  this  section,  but  for 
the  first  examples  in  all  the  sections. 

The  pupil  will  probably  solve  the  first  examples 
without  any  instruction. 

Examples  in  addition  and  subtraction  may  be 
solved  by  Plate  I.  as  follows : — 

How  many  are  5  and  3  ?*  Select  a  rectangle  con-» 
taming  5  marks,  and  another  containing  3  marks, 
and  ascertain  the  number  of  marks  in  both. 

How  many  are  8  and  6  ?  Select  a  rectangle  contain- 

*  Figures  are  used  in  the  Key,  because  the  instructer  is  supposed  to 
be  acquainted  with  them.     They  are  not  used  in  the  first  part  of  tf  ^ 
book,  because  the  pupil  would  not  understand  them  so  well  as  he  v 
the  words. 


SECT.  1.]  KEY.  145 

ing  8  marks,  and  another  containing  6  marks,  and 
count  them  together. 

How  many  are  17  and  5  ?  Keeping  17  in  the 
mind,  select  a  rectangle  containing  5  marks,  and 
add  them  thus  : — 17  and  1  are  18,  and  1  are  19,  and 
1  are  20,  and  1  are  21,  and  1  are  22. 

If  you  take  4  from  9,  how  many  will  remain  ? 
Select  a  rectangle  containing  9  marks,  and  take 
away  four  of  them. 

18  less  5  are  how  many  ?  Keeping  18  in  mind, 
select  a  rectangle  containing  5,  and  take  them  away 
1  at  a  time. 

In  this  manner  all  the  examples  in  this  section 
may  be  solved. 

B  &  C.  The  articles  B  and  C  contain  the  com- 
mon addition  table  as  far  as  the  first  10  numbers.  In 
the  first  the  numbers  are  placed  in  order,  and  in 
the  second  out  of  order. 

The  pupil  should  study  these  until  he  can  find  the 
answers  readily,  and  then  he  should  commit  the 
answers  to  memory. 

D.  In  this  article,  the  numbers  are  larger  than  in 
the  preceding,  and,  in  some  instances,  three  or  more 
numbers  are  added  together.  In  the  abstract  ex- 
amples, the  numbers  from  one  to  ten  are  to  be  added 
to  the  numbers  from  ten  to  twenty. 

4E.   This  article  contains  subtraction. 
_j_ A 

F.  This  article  is  intended  to  make  the  pupil 
familiar  with  adding  the  nine  first  numbers  to  all 
others.  The  pupil  should  study  it  until  he,  can  an- 
swer the  questions  very  readily. 

€r.  In  this  article,  all  the  preceding  are  combined 
13 


145  KEY.  [PART  2. 

together,  and  the  numbers  from  1  to  10  are  added 
to  all  numbers  from  20  to  100;  and  subtracted  in 
the  same  manner. 

18.  f>7  and  <>  arc  03,  and  3  are  06,  and  5  are  71, 
^and  2  are  73,  less  8  are  65. 

H.  This  article  contains  practical  questions  which 
show  the  application  of  all  the  preceding  articles. 

0.  37  less  5  are  32,  less  8  are  24,  less  6  (which 
he  kept  himself)  are  18 ;  consequently  he  gave  18 
to  the  third  boy. 


SECTION  II. 

THIS  section  contains  multiplication.     The  pupil 
will   see  no  difference   between  this  and  addition. . 
It  is  best  that  he  should  not  at  first,  though  it  may 
be  well  to'explain  it  to  him  after  a  while. 

A.  This  article  contains  practical  questions,  which 
the  pupil  will  readily  answer. 

I.  Three  yards  will  cost  3  times  as  much  as   1 
yard. 

N.  B.  Be  careful  to  make  the  pupil  give  a  similar 
reason  for  multiplication,  both  in  this  article,  and 
elsewhere. 

This  question  is  solved  on  the  plate  thus  :  in  the 
second  row,  count  3  rectangles,  and  find  their  sum, 
2  and  2  are  4,  and  2  are  0. 

II.  A  man  will  travel  4  times  as  far  in  4  hours  as 
he  will  in  1  hour.     In  the  third  row,  count  4  times 
3,  and  ascertain  their  sum. 

15.  There  are  4  times  as  many  feet  iu  4  yards  as 
in  1  yard,  or  4  times  3  feet. 


SECT.  2.]  KEY.  147 

B.  This  article  contains  the  common  multiplica- 
tion  table,  as  far  as  the   product  of  the   first  ten 
numbers.     The  pupil  should  find  the  answers  once 
or  twice  through,  until  he  can  find  them  readily, 
and  then  let  him  commit  them  to  memory. 

^43.  6  times  3.     In  the  third  row  count  6  times 

3,  and  then  ascertain  their  sum.    3  and  3  are  6,  &,c. 

59.  7  times  9.     In  the  ninth  row  count  7  times 

9,  or  7  rectangles,  and  ascertain  their  sum.     9  and 

9  are  18,  &c. 

C.  This  article  is  the  same  as  the  preceding,  ex- 
cept in  this  the  numbers  are  out  of  their  natural 
order. 

I>.  In  this  article  multiplication  is  applied  to  prac- 
tical  examples.      They   are  of  the   same   kind   as 
those  in  article  A  of  this  section. 
•  12.  There  are  8  times  as  many  squares  in  8  rows 
as  in  1  row.     8  times  8  are  64. 

13.  There  are  6  times  as  many  farthings  in  6 
pence  as  in  1  penny.  6  times  4  are  24. 

17.  12  times  4  are  48. 

Note.  When  a  number  is  taken  more  than  10 
times,  as  in  the  above  example,  after  taking  it  10 
times  on  the  plate,  begin  at  the  beginning  of  the 
row  again,  and  take  enough  to  make  up  the  number. 

23.  There  are  3  times  as  many  pints  in  3  quarts 
as  in  1  quart.  3  times  2  are  6.  And  in  6  pints 
there  are  6  times  4  gills,  or  24  gills. 

28.  In  3  gallons  there  are  12  quarts,  and  in  12 
quarts  there  are  24  pints. 

31.  In  2  gallons  are  8  quarts,  in  8  quarts  16  pints, 
in  16  pints  64  gills.  16  times  4  are  64. 

35.  In    1   gallon   are  32   gills ;  and  32   times  2 


148  KEY.  [PART  2. 

cents  are  64  cents.     Or,  1  pint  will  cost  8  cents, 
and  there  are  8  pints  in  a  gallon.     8  times  8  are  (>4. 
38.  They  will  be  2  miles  apart  in  one  hour,  4  miles 
in  2  hours,  &c. 


SECTION  III. 

A.  THIS  section  contains  division.      The  pupil 
will  scarcely  distinguish  it  from  multiplication.     It 
is  not  important  that  he  should  at  first. 

Though  the  pupil  will  be  able  to  answer  these 
questions  by  the  multiplication  table,  if  he  has  com- 
mitted it  to  memory  thoroughly ;  yet  it  will  be  bet- 
ter to  use  the  plate  for  some  time. 

9.  As  many  times  as  3  dollars  are  contained  in 
15  dollars,  so  many  yards  of  cloth  may  be  bought 
for  15  dollars.  On  Plate  I.,  in  the  third  row,  count 
fifteen,  and  see  how  many  times  3  it  makes.  It  is 
performed  very  nearly  like  multiplication. 

B.  In  this  article  the  pupil  obtains  the  first  ideas 
of  fractions,  and  learns  the  most  important  of  the 
terms  which  are  applied  to  fractions.*     The  pupil 
has  already  been  accustomed  to  look  upon  a  collec- 
tion of  units,  as  forming  a  number,  or  as  being  itself 
a  part  of  another  number.     He   knows,  therefore, 
that  one  is  a  part  of  every  number,  and  that  every 
number  is  a  part  of  every  number  larger  than  itself. 
As  every  number  may  have  a  variety  of  parts,  it  is 
necessary  to  give  names  to  the  different  parts,  in  order 
to  distinguish  them   from  each  other.     The  parts 

*  As  soon  as  the  terms  applied  to  fractions  are  fully  comprehended, 
the  operations  on  them  are  as  simple  as  those  on  whole  numbers. 


SECT.  3.]  KEY.  149 

receive  their  names,  according  to  the  number  of 
parts  which  any  number  is  divided  into.  If  the 
number  is  divided  into  two  equal  parts,  the  parts  are 
called  halves;  if  it  is  divided  into  three  equal  parts, 
they  are  called  thirds ;  if  into  four  parts,  fourths, 
&/c. ;  and,  having  divided  a  number  into  parts,  we 
can  take  as  many  of  the  parts  as  we  choose.  If  a 
number  be  divided  into  five  equal  parts,  and  three 
of  the  parts  be  taken,  the  fraction  is  called  three 
fifths  of  the  number.  The  name  shows  at  once  into 
how  many  parts  the  number  is  to  be  divided,  and 
how  many  parts  are  taken. 

The  examples  in  this  book  are  so  arranged,  that 
the  names  will  usually  show  the  pupil  how  the  ope- 
ration is  to  be  performed.  In  this  section,  although 
the  pupil  is  taught  to  divide  numbers  into  various 
parts,  he  is  not  taught  to  notice  any  fractions,  ex- 
cept those  where  the  numbers  are  divided  into  their 
simple  units,  which  is  the  most  simple  kind. 

It  will  be  best  to  use  beans,  pebbles,  &,c.  first ; 
and  then  Plate  I. 

4.  Show  the  pupil  one  of  the  rectangles  in  the 
second  row,  and  explain  to  him  that  one  is  1  half 
of  2. 

7.  In  the  second  row  count  3  units ;  it  will  take 
all  the  marks  in  the  first,  and  1  in  the  second  rec- 
tangle. Consequently  it  is  1  time  2,  and  1  half  of 
another  2. 

15.  In  the  second  row  count  9.  It  will  take  all 
the  marks  in  the  four  first  rectangles,  and  1  in  the 
fifth.  Therefore  9  is  4  times  2  and  one  half  of 
another  2. 

18.  Show  the  pupil  a  rectangle  in  the  third  row, 
and  ask  him  the  question,  and  explain  to  him  that  1 
is  1  third  of  3. 

20.  Since  1  is  1  third  of  3,  2  must  be  2  thirds  of  3. 

34.  In  the  third  row  count  11.  It  will  take  3 
13* 


150  KEY.  [PART  2. 

rectangles  and  2  marks  in  the  fourth.  Therefore 
11  is  3  times  3,  and  2  thirds  of  another  3. 

Proceed  in  the  same  manner  with  the  other  di- 
visions. 

This  being  one  of  the  most  useful  combinations, 
and  one  but  very  little  understood  by  most  people, 
especially  when  applied  to  large  numbers,  the  pu- 
pil must  be  made  perfectly  familiar  with  it.  Ask 
questions  like  those  in  the  book  for  large  numbers, 
and  also  some  like  the  following :  What  part  of  7 
is  18  ?  The  answer  will  be  If-. 

C.  The  first  ten  figures  are  here  explained. 
They  are  used  as  an  abridged  method  of  writing 
numbers,  and  not  with  any  reference  to  their  use  in 
calculating. 

This  article  is  only  a  continuation  of  the  last. 
All  the  numbers  from  1  to  100  are  introduced  into 
the  two  articles,  and  are  divided  by  all  the  numbers 
from  1  to  10  ;  except  that  some  of  the  largest  are 
not  divided  by  some  of  the  smallest. 

2.  The  pupil  answers  first,  how  many  times  2  is 
contained  in  12,  then  how  many  times  3. 

45.  63  are  how  many  times  5?  In  the  fifth  row 
count  63.  It  will  take  12  rectangles  and  3  marks 
in  the  13th.  It  will  be  necessary  to  count  once 
across  the  plate,  and  begin  again,  and  take  2  rec- 
tangles and  a  part  of  the  third.  63  is  12  times  5 
and  3  fifths  of  another  5. 

D.  These  examples,  which  are  similar  to  those 
in  article  A  of  this  section,  are  solved  in  the  same 
manner. 

5.  It  would  take  as  many  hours  as  3  miles  are 
contained  in  10  miles.  3  hours  and  £  of  an  hour. 

20.  They  cost  as  many  cents  as  there  are  3  ap- 
ples in  30  apples  ;  that  is,  10  cents. 


SECT.  4.]  KEY.  151 

21.  12  dollars  a  month :  and  12  dollars  a  month 
is  3  dollars  a  week  ;  that  is,  18  shillings  a  week, 
which  is  3  shillings  a  day. 

26.  The  whole  loss  was  35  dollars,  which  was  7 
dollars  apiece. 


SECTION  IV. 

A.  THIS  article  contains  multiplication  simply. 
It  is  repeating  a  number  a  certain  number  of  times 
and  a  part  of  another  time. 

14.  6  times  5  are  30,  and  -|  of  5  are  3,  which 
added  to  30  make  33.  On  the  plate  in  the  fifth 
row,  take  6  rectangles  and  3  marks  in  the  seventh, 
and  ascertain  their  sum. 

B.  In  this  article  the  pupil  is  taught  to  change  a 
certain  number  of  twos  into  threes,  threes  into  fives, 
&c.     This  article  combines  all  the  preceding  oper- 
ations. 

24.  4  cords  of  wood  will  cost  28  dollars,  and  f  of 
a  cord  will  cost  2  dollars,  which  makes  30  dollars. 
30  dollars  will  buy  3  hundred  weight  of  sugar  and 
f  of  another  hundred  weight. 

29.  7  times  8  are  56,  and  f  of  8  are  5,  which 
added  to  56  make  61 ;  61  are  6  times  9,  and  £  of  9. 

C.  1.  4  bushels  of  apples,  at  3  shillings  a  bushel, 
come  to  12  shillings ;    and  12  shillings  are  2  dollars. 

2.  The  2  lemons  come  to  8  cents,  and  8  cents 
will  buy  4  apples,  at  2  cents  apiece. 

This  is  usually  called  Barter.  The  general  prin- 
ciple is  to  find  what  the  article  will  come  to,  whose 
price  and  quantity  are  given,  and  then  to  find  how 
much  of  the  other  article  that  money  will  buy. 


152  KEY.  [PART  2. 

6.  If  2  apples  cost  4  cents,  1  will  cost  2  cents, 
and  4  will  cost  8  cents.  Or  4  apples  will  cost  2 
times  as  much  as  2  apples. 

22.  Find  how  many  times  2  pears  are  contained 
in  20  pears,  which  is  10  times.     10  times  3  cents 
are  30  cents.     Or,  first  find  what  20  pears  would 
come  to,  at  3  cents  apiece ;  and  since  it  is  2  for  3 
cents,  instead  of  1  for  3  cents,  the  price  will   be 
half  as  much. 

23.  See  how  many  times  you  can  have  5  cents 
in  30  cents,  and  you  can  buy  so  many  times  3  eggs. 
30  is  6  times  5,  and  6  times  3  are  18.     18  eggs. 

24.  10  dollars  a  week,  and  40  dollars  a  month. 

25.  5  dollars  are  30  shillings,  which  is  10  shil- 
lings a  day. 

26.  5  dollars  apiece. 


SECTION  V. 

IN  this  section  the  principle  of  fractions  is  applied 
to  larger  numbers,  but  such  as  are  divisible  into  the 
parts  proposed  to  be  taken.  The  pupil  who  is  fa- 
miliar with  what  precedes,  will  easily  understand 
the  examples  in  this  section.  They  require  nothing 
but  division  and  multiplication. 

A.  Let  the  pupil  explain  each  example  in  the 
following  manner : — What  is  1  sixth  of  18  ?  Ans.  3. 
Why  ?  Because  6  times  3  are  18 ;  therefore  if  you 
divide  18  into  6  equal  parts,  one  of  the  parts  will 
be  3. 

To  find  this  answer  on  the  plate  ;  on  the  6th  row, 
the  pupil  will  find  3  times  6  make  18;  this  will 
direct  him  to  the  third  row,  where  he  will  find  6 


SECT.  5.}  KEY.  153 

times  3  are  18.  Consequently,  he  will  see  18  divided 
into  6  equal  parts.  It  will  be  well  to  let  the  pupil 
prove  a  large  number  of  the  examples  on  the  plate. 

The  pupil  will  be  very  likely  to  say,  3  is  the  6th 
part  of  18,  because  3  times  6  are  18.  Be  careful  to 
make  him  say  it  the  other  way,  viz.  6  times  3  are  18. 

14.  1  third  of  9  is  3 ;  f  is  2  times  as  much  as  £, 
therefore  f  of  9  is  6. 

19.  1  barrel  will  cost  %  part  of  12  dollars  ;  3  bar- 
rels will  cost  |  of  12  dollars.  7  barrels  will  cost  J 
of  12  dollars. 

37.  What  is  $  of  32  ?  $  of  32  is  4,  f  are  5  times 
4,  or  20. 

B.  11.  |  of  20  is  4 ;  £  are  7  times  4,  or  28 ;  and 
28  is  4  times  6,  and  ^  of  6. 

C.  3.  1  half  of  10  is  5,  f  of  10  are  4 ;  5  and  4 

are  9.     He  gave  away  nine,  and  had  1  left. 

4.  1  yard  will  cost  £  of  what  3  yards  cost.     £  of 
6  dollars  is  2  dollars. 

5.  2  yards  will  cost  1  half  of  what  4  cost ;  or  6 
dollars. 

6.  3  apples  will  cost  ^  of  what  9  cost ;  or  6  cents. 

7.  2  is  §  of  3  ;  therefore  2  oranges  will  cost  f  of 
what  3  cost,     f  of  18  cents  are  12  cents. 

8.  |  of  25  are  20.     The  10  apples  cost  20  cents, 
which  was  2  cents  apiece. 

11.  f  of  42  are  12,  and  6  times  12  are  72.  72 
dollars. 

13.  3  is  f  of  4.     J  of  12  dollars  are  9  dollars. 
Or,  4  yards  at  12  dollars  is  3  dollars  a  yard,  and  9 
dollars  for  3  yards. 

14.  Solved  like  the  13th.     Ans.  15  cents. 

15.  Since  1  is  $  of  3,  7  is  J  of  3.     £  of  15  cents 
are  35  cents.    Or,  3  oranges  at  15  cents  is  5  cents 
apiece  :  7  times  5  are  35  cents. 


154  KEY.  [PART  2. 

Note.  In  questions  of  this  kind,  it  is  generally  the 
simplest  way  to  find  what  1  article  will  cost ;  then  it 
may  easily  be  told  how  much  any  number  will  cost. 

19.  4  men  would  do  it  in  1  half  the  time  that  2 
would  do  it.  Or,  you  may  say,  If  2  men  would  do 
it  in  6  days,  1  man  would  do  it  in  12  days,  and  4 
men  in  £  of  that  time,  or  three  days. 


SECTION  VI. 

A.  4.  2  halves  of  any  number  make  the  whole 
number.     Therefore  2  is  1  half  of  2  times  2,  or  4. 
It  is  £  of  4  times  2,  or  8. 

Let  the  pupil  answer  these  questions  in  the  fol- 
lowing manner  : — 4  is  -J-  of  3  times  4 ;  3  times  4  are 
12.  5  is  ij-  of  7  times  5 ;  7  times  5  are  35. 

B.  2.  4  is  2  times  2. 
4.  6  is  2  times  3. 

16.  2  thirds  of  any  number  is  twice  as  much  as  $ 
of  the  same  number.  If  4  is  §  of  some  number, 
then  1  half  of  4,  or  2,  is  £  of  that  number  :  2  is  £  of 
6 ;  therefore  4  is  f  of  6. 

20.  If  6  is  f  of  a  number,  £  of  6,  or  2,  is  |  of  the 
same  number :  2  is  £  of  8  ;  therefore  6  is  f  of  8. 

23.  It  is  evident  that  $  of  a  pound  will  cost  only 
•J  of  what  £  will  cost.  If  f  cost  6  cents,  -f  will  cost 
2  cents,  and  the  whole  pound  14  cents. 

26.  It  will  probably  be  perceived,  by  this  time, 
that,  ^  of  a  number  being  given,  it  is  necessary  to 
find  ^-,  and  then  the  number  is  easily  found ;  4  be- 
ing f ,  2  is  |,  and  2  is  |  of  14. 

45.  24  being  f ,  £  of  24  or  3  will  be  | ;  3  is  £  of  27. 


SECT.  6.]  KEY.  155 

C.  6:  20  being  f,  5  is  |,  and  5  is  |  of  35 ;  and  35 
is  5  times  6,  and  %  of  6. 

I>.  4.  18  is  3  times  6,  and  6  is  £  of  4  times  6,  or 
24.  Ans.  24  dollars. 

6.  54  is  |  of  48 ;  12  yards  at  48  dollars  is  4  dol- 
lars a  yard.     He  gained  6  dollars. 

7.  10  feet  is  f  of  15  feet. 

8.  If  f  are  under  water,  there  must  be  -J  out  of 
the  water.     4  is  £  of  12. 

9.  If  f  are  under  water,  there  must  be  f  out  of 
the  water.     6  is  f  of  10. 

10.  |  and  f  are  -f .     fy  bear  cherries  and  peaches ; 
consequently,  the  10  which  bear  plums  must  be  the 
other  f ;  10  is  f  of  35.     10  bear  peaches,  and  15 
bear  cherries. 

11.  f,  and  f,  and  £,  and  £,  are  £;  therefore  12 
must  be  the  other  f  of  the   whole.     The   whole 
number  is  54. 

Miscellaneous  Examples. 

6.  The  ^greyhound  gains  upon  the  fox  4  rods  in 
a  minute.  It  will  take  him  20  minutes  to  gain  80 
rods. 

8.  f  of  24.     Or  you  may  say,  1  sheep  would  cost 
3  dollars,  and  3  sheep  9  dollars. 

9.  30  horses  will  eat   10  times   as  much  as  3 
horses. 

11.  10  dollars  apiece,  and  2  dollars  a  yard. 

12.  5  dollars  for  1  week,  20  dollars  for  a  month, 
and  25  dollars  for  5  weeks. 

14.  It  would  take  them  5  times  as  long  to  eat  40 
bushels,  as  it  would  to  eat  8  bushels. 

15.  4  horses  would  eat  4  bushels  in  3  days,  and 
it  would  take  them  9  times  as  long  to  eat  36  bush- 
els.    Ans.  27  days. 


156  KEY.  [PART  2. 

16.  If  2  men  spend  12  dollars  in  1  week,  1  man 
will  spend  6  dollars  in  1  week,  and  30  dollars  in  5 
weeks,  and  3  men  would  spend  3  times  as  much,  or 
90  dollars. 

17.  The  shadow  of  the  staff  is  §  of  the  length 
of  the  staff;  therefore  the  shadow  of  the  pole  must 
be  f  the  length  of  the  pole.     18  feet  is  \  of  27 
feet. 

20.  It  would  take  2  men  3  times  as  long  to  do  it 
as  it  would  6  men. 

23.  8  men  would  do  a  piece  of  work  1  half  as 
large  in  2  days,  and  it  would  take  2  men  4  times  as 
long  to  do  it,  or  8  days. 

28.  He  must  sell  it  for  56  dollars  in  order  to  gain 
16  dollars.     56  dollars  is  7  dollars  per  barrel. 

29.  It  cost  him  35  dollars,  and  he  must  sell  it  for 
45  to  gain  10  dollars ;  45  dollars  is  9  dollars  a  firkin. 

30.  Ans.  56  cents.— See  Section  VI. 

33.  If  it  would  last  3  men  10  months,  it  would 
last  1  man  30  months,  and  5  men  6  montjis. 

34.  There  are  8  times  5  in  40 ;  and  since  the 
other  would  build  as  many  times  9,  as  the  first  does 
5,  he  would  build  8  times  9,  or  72  rods. 


SECTION  VII. 

A.  13.  £of  20is4,f  are  16;  16  being  f,  2  isj; 
2  is  |  of  14,  and  16  is  f  of  14. 

16.  f  of  28  are  12 ;  12  is  2  times  6,  and  6  is  £  of 

48,  (12  is  f  of  48)  and  48  is  6  times  7  and  f  of  7. 

B.  1.  ±  of  15  are  12;  12  is  6  times  2;  2  is  ^ 
of  20  (12  is  ft  of  20) ;  £  of  21  is  7  ;  20  is  2  times 
7  and  f.  of  7. 


SECT.  8.]  KEY.  157 

2.  |  of  18  are  24;  24  is  f  of  27;  |  of  35  is  5; 
27  is  5  times  5  and  f  of  5. 

C.  This  article  contains  the  multiplication  table, 
in  which  the  numbers  from  10  to  20  are  multiplied 
by  the  ten  first  numbers. 


SECTION  VIII. 

Explanation  of  Plate  II. 

PLATE  I.,  which  has  been  used  in  the  preceding 
sections,  presents  each  unit  as  a  simple  object,  and 
undivided.  Plate  II.  presents  the  units  as  divisible 
objects,  the  different  fractions  of  which  form  parts, 
and  sums  of  parts,  of  unity. 

This  plate  is  divided  into  ten  rows  of  equal 
squares,  and  each  row  into  ten  squares. 

The  first  row  is  composed  of  ten  empty  squares, 
which  are  to  be  represented  to  the  pupil  as  entire 
units.  The  second  row  presents  ten  squares,  each 
divided  into  two  equal  parts  by  a  vertical  line ;  each 
of  these  parts,  of  course,  represents  one  half.  In 
the  third  row,  each  square  is  divided  into  three 
equal  parts,  by  two  vertical  lines,  each  part  repre- 
senting one  third,  &c.,  to  the  tenth  row,  which  is  di- 
vided into  ten  equal  parts,  each  part  representing 
one  tenth  of  unity. 

N.  B.  In  Plates  II.  and  III.,  the  spaces,  and  not 
the  marks,  are  to  be  counted. 

Be  careful  to  make  the  pupil  understand,  1st, 
that  each  square  on  the  plate  is  to  be  considered  as 
14 


158  KEY.  [PART  2. 

an  entire  unit,  or  whole  one.  2d,  explain  the  divi- 
sions into  two,  three,  four,  &,c.  parts.  3d,  teach  him 
to  name  the  different  parts.  Make  him  observe  thv-it 
the  name  shows  into  how  many  parts  one  is  divided, 
and  how  many  parts  are  taken,  in  the  same  manner 
as  it  does  when  applied  to  larger  numbers.  4,  for 
example,  shows  that  one  thing  is  to  be  divided  into 
7  equal  parts,  and  4  of  those  parts  are  to  be  taken. 
4th,  make  the  pupil  compare  the  different  parts  to- 
gether, and  observe  which  is  the  largest.  Ask  him 
such  questions  as  the  following :  Which  are  the 
smaller,  halves  or  thirds?  Ans.  Thirds.  Why? 
Because  the  more  parts  a  thing  is  divided  into,  the 
smaller  the  parts  must  be. 

A.  15.  On  Plate  II.,  count  two  squares  in  the 
second  row,  and  then  ascertain  the  number  of  spaces 
or  halves  in  them.  There  are  4  halves. 

21.  In  the  2d  row,  take  3  squares  and  1  space  in 
the  4th  square;  then  count  the  spaces.  Ans.  7 
halves. 

37.  In  the  third  row,  take  5  squares,  and  2  spaces 
in  the  6th  ;  then  count  the  spaces  or  thirds.      Ans. 
17  thirds. 

54.  In  the  5th  row,  take  6  squares,  and  4  spaces 
in  the  7th  square ;  then  count  the  spaces  or  fifths. 
Ans.  34  fifths. 

H.  2.  This  operation  is  the  reverse  of  the  last. 
In  the  2d  row,  count  4  spaces  or  halves,  and  see  how 
many  squares  or  whole  ones  it  takes.  It  will  take  2. 

38.  In  the  9th  row,  count  48  spaces  or  9ths,  and 
see  how  many  squares  or  whole  ones  it  takes.     It 
will  take  5  squares  and  3  spaces  in   the  6th.     Ans. 
5  whole  ones  and  £. 


SECT.  9.]  KEY.  159 

SECTION  IX. 

A.  2.  £  signifies  that  1  thing  is  divided  into  3 
equal  parts,  and  1  part  taken.  Therefore  2  times 
1  third  is  2  parts,  or  f . 

6.  7  times  £  is  J,  or  2£. 

10.  On  the  plate  in  the  3d  row,  5  times  f  are  J^0-, 
which  takes  3  squares  and  1  space.  Ans.  3£. 

24.  In  the  9th  row,  take  4  spaces  or  9ths,  and 
repeat  them  5  times,  which  will  make  %°-,  and  will 
take  2  squares  and  2  spaces.  Ans.  2f . 

15.  4.  4  times  2  are  8,  and  4  times  1  half  are  4 
halves,  or  2,  which  added  to  8  make  10. 

18.  4  times  3  are  12,  and  4  times  f  are  *£-,  or 
three  whole  ones,  which  added  to  12  make  15. 

32.  2  times  3  are  6,  and  2  times  ^  are  f ,  which 
added  to  6  make  6^. 

40.  10  barrels  of  cider  at  3  dollars  and  |  a  bar- 
rel ;  10  barrels  at  3  dollars  would  be  30  dollars, 
then  10  times  £  is  -5^,  or  8  and  f  of  a  dollar.  Ans. 
38f  dollars. 

C.  2.  J  to  each  would  be  3  times  f ,  or  f ,  which 
are  2£  oranges. 

3.  J£.  or  2  bushels. 

4.  7  times  j-  are  -^L,  or  5£  gallons. 

5.  8  yards  and  f  or  2  yards,  that  is,  10  yards. 

6.  4  times  2  are  8,  and  4  times  -f  are  -^,  or  2f , 
which  added  to  8  make  10f  bushels. 

12.  It  would  take  1  man  3  times  as  long  as  it 
would  3  men.  Ans.  13f  days. 

14.  3  men  would  build  3  times  as  much  as  1  man  ; 
and  in  4  days  they  would  build  4  times  as  much  as 
in  1  day.     ^.715.  38f  rods. 

15.  Ans.  12  yards. 


360  KEY.  [PART  2. 

SECTION  X. 

A.  21.  £of  1  is  £.   £  of  2  is  2  times  as  much,  or 
f .     i  of  4  is  f,  or  If     i  of  5  is  f ,  or  If.     £  of  6 
is  f,  or  2.     |  of  7  is  J,  or  2f 

27.  £  of  1  is  f  |  of  2  is  f .  £  of  3  is  f .  £  of 
7  is  £,  or  If 

This  manner  of  reasoning  may  be  applied  to  any 
number.  To  find  J  of  38  :  it  is  Sf-,  for  |  of  38  is  38 
times  as  much  as  j-  of  1,  and  ^  of  1  is  | ;  consequent- 
ly |  of  38  is  ^8.,  and  $p.  is  5f 

40.  To  find  f  of  a  number,  £  must  be  found  first, 
and  then  f  will  be  2  times  as  much.  -J  of  7  is  J, 
and  2  times  J  are  4^-,  or  4f . 

74.  £  of  50  is  ^°-,  or  5| ;  f  is  4  times  as  much ; 
4  times  5  are  20,  4  times  %  are  ^,  or  2f ,  which 
added  to  20  make  22f . 

NOTE.  The  manner  employed  in  example  40th  is  best  for  small 
numbers,  and  that  in  the  74th  for  large  numbers. 

B.  2.  Am.  If  apiece. 

3.  £  of  3  is  f ;  f  of  a  bushel  apiece. 

4.  f  of  7  is  4^ ;  he  gave  away  4|,  and  kept  2|. 

6.  1  half  dollar  a  yard,  or  50  cents. 

7.  i  of  7  is  I,  or  If ;  f  of  a  dollar  is  f  of  100 
cents,  which  is  40  cents.  Ans.  1  dollar  and  40  cents 
a  bushel. 

8.  |  of  8  is  If.     f  of  100  is  33f .     Ans.  1  dollar 
and  33f  cents ;  or  it  is  1  dollar  and  2  shillings. 

9.  If  3  bushels  cost  8  dollars,  1  bushel  will  cost 
2  dollars  and  f,  and  2  bushels  will  cost  5£  dollars. 
Ans.  5  dollars  and  2  shillings,  or  33f  cents. 

13.  If  7  pounds  cost  40  cents,  1  will  cost  5|-  cents  ; 
10  pounds  will  cost  57|  cents. 

16.  1  cock  would  empty  it  in  6  hours,  and  7  cocks 


SECT.  11.]  KEY.  161 

would  empty  it  in  |  of  6  hours,  or  f.  of  1  hour, 
which  is  f-  of  60  minutes ;  ^  of  60  minutes  is 
minutes. 


SECTION  XL 

A,  2.  2  halves  of  a  number  make  the  number ; 

consequently  1  and  1  half  is  the  half  of  2  times  1 

and  1  half,  which  is  3. 

15.  4f.  is  |  of  5  times  4  and  f ,  which  is 
17.  4$  is  |  of  9  times  4f ,  which  is  39f 

IB.  4.  5  is  3  times  %  of  5,  which  is  -J,  or  If. 

30.  If  8  is  |  of  some  number,  $  of  S  is  £  of  the 
same  number.  %  of  8  is  2§ ;  2f  is  £  of  4  times  2f , 
which  is  lOf ;  therefore  8  is  £  of  lOf. 

40.  If8isf  iofSisf;  *of8isf;  fis|of^, 
or  9f ;  therefore  8  is  f  of  9f. 

52.  If  |^  of  a  ton  cost  23  dollars,  £  of  a  ton  must 
be  -£  of  23,  that  is,  4f  dollars,  and  the  whole  would 
cost  9  times  as  much,  that  is,  41f . 

69.  £  of  65  is  7f :  7f  is  |  of  5  times  7f ,  which  is 
38£.  65  is  f  of  36f 

C.  4.  37  is  f  of  32f ,  which  taken  from  37  leaves 
4£.  Ans.  4^-  dollars. 

5.  7  feet  must  be  f  of  the  whole  pole. 

6.  If  he  lost  f ,  he  must  have  sold  it  for  £  of  what 
it  cost.     47  is  £  of  60f .     Ans.  60  dollars  and  42f 
cents. 

Miscellaneous  Examples. 

1.  The  shadow  of  the  staff  is  f  of  the  length  of 
the  staff;  therefore  the  shadow  of  the  pole  is  %  of 
the  length  of  the  pole.     67  is  £  of  83f .     Ans.   83f 
feet. 

2.  9  gallons  remain  in  the  cistern  in  1  hour.    It 
will  be  filled  in  10  hours  and  £ ;  £  of  60  minutes 

14* 


163  KEY.  [PART  2. 

are  46  minutes  and  f ;  f  of  60  seconds  are  40  sec- 
onds.    Ans.  10  hours,  46  minutes,  40  seconds. 

10.  Find  f  of  33,  and  subtract  it  from  17.  Ans.  3£. 

11.  It  will  take  3  times  10  yards. 

13.  5  is  £  of  3 ;  it  will  take  f  as  much.  Or,  7 
yards,  5  quarters  wide,  are  equal  to  35  yards  1  quar- 
ter wide,  which  is  equal  to  llf  yards  that  is  3 
quarters  wide. 

15.  £  of  37  dollars. 

16.  £  as  much. 


SECTION  XII. 

THE  examples  in  this  section  are  performed  in 
precisely  the  same  manner  as  those  in  the  sections 
to  which  they  refer.  All  the  difficulty  consists  in 
comprehending  that  fractions  expressed  in  figures 
signify  the  same  thing  as  when  expressed  in  words. 
Make  the  pupil  express  them  in  words,  and  all  the 
difficulty  will  vanish.  Let  particular  attention  be 
paid  to  the  explanation  of  fractions  given  in  the 
section. 

VIII.  A*  6.  In  7  how  many  £  ?     expressed  in 
words,  is,  In  7  how  many  sixths  ?  Ans.  *•£-. 

14.  Reduce  Sy3^  to  an  improper  fraction  ;  that  is, 
In  8  and  3  tenths,  how  many  tenths  ?  Ans.  f§. 

B.  8.  %f-  are  how  many  times  1  ?  that  is,  In  23 
sevenths  how  many  whole  ones?  Ans.  3^' 

IX.  B.  3.  How  much  is  5  times  6f  ?    that  is, 
How  much  is  5  times  6  and  4  sevenths  ?    Ans.  32f . 

V.  &,  X.  15.  What  is  f  of  27?  that  is,  What  is 
5  eighths  of  27?  Ans.  16|. 

VI.  &  XI.  A.  8.  7f  is  £  of  what  number?  that 
is,  7  and  6  sevenths  is  1  eighth  of  what  number  ? 
Ans.  62f . 


SECT.  Ifc]  KEY.  163 

B.  4.  12  is  f  of  what  number  ?  that  is,  12  is  3 
sevenths  of  what  number  ?  Ans.  28. 

12.  4  is  £  of  what  number  ?  that  is,  4  is  3  fifths  of 
what  number  ?  Ans.  6f . 


Explanation  of  Plate  III. 

Plate  III.  is  intended  to  represent  fractions  of 
unity,  divided  into  other  fractions ;  it  is,  therefore, 
an  extension  of  Plate  II.  It  differs  from  it  only 
in  this,  that,  besides  the  vertical  divisions,  the 
squares  are  divided  horizontally,  so  as  to  cut  the 
fractions  of  the  square  into  fractions  of  fractions. 
The  horizontal  lines  are  dotted,  but  they  are  to  be 
considered  as  lines. 

This  plate,  like  the  preceding,  is  divided  into 
ten  rows  of  squares,  each  row  containing  ten  equal 
squares.  In  the  first  row,  the  first  square  is  undi- 
vided ;  the  9  following  squares  are  divided  by  hori- 
zontal lines  into  from  two  to  ten  equal  parts.  In  all 
the  other  squares,  the  vertical  divisions  are  the  same 
as  in  Plate  II. ;  and,  besides  this,  each  row  is  divided 
horizontally  in  the  same  manner  as  the  first  row. 

By  means  of  this  double  division,  the  second  row 
presents  a  series  of  fractions,  from  halves  to  twen- 
tieths. The  3d  row  presents  a  series  from  thirds 
to  thirtieths,  and  so  on  to  the  tenth  row,  which 
presents  a  series  from  tenths  to  hundredths. 

The  2d  row,  besides  presenting  halves,  fourths, 
sixths,  eighths,  &c.,  shows  also  halves  of  halves, 
thirds  of  halves,  fourths  of  halves,  &c.,  and  shows 
their  ratios  with  unity'. 

The  3d  row,  besides  thirds,  sixths,  ninths,  &/c., 
shows  halves  of  thirds,  thirds  of  thirds,  &c.,  and 
their  ratios  with  unity.  The  other  rows  present 
analogous  divisions. 


164  KEY.  [PART  2. 

SECTION  XIII. 

THE  operations  in  this  section  are  the  reducing 
of  fractions  to  a  common  denominator,  and  the  ad- 
dition and  subtraction  of  fractions.  The  examples 
will  generally  show  what  is  to  be  done,  and  how  it 
is  to  be  done.  Plate  III.  will  be  found  very  useful 
in  explaining  the  operations,  by  exhibiting  the  di- 
visions to  the  eye. 

1.  The  first  example  may  be  illustrated  by  the 
second  square  in  the  second  row.  This  square  is 
divided  into  halves  by  a  vertical  line,  and  then  into 
fourths  by  the  horizontal  line.  It  will  be  readily 
seen  that  £  makes  ,2  fourths,  and  that  the  first  had 
twice  as  much  as  the  second.  The  plate  will  not 
be  so  necessary  for  the  practical  questions  as  for 
the  abstract.  In  the  second  example,  therefore,  it 
will  be  more  useful  than  in  the  first. 

4.  It  will  readily  be  seen  on  the  second  square  of 
the  second  row,  that  £  and  £  are  £. 

8.  It  will  be  seen  in  the  third  square  of  the  sec- 
ond row,  that  £  makes  f . 

10  and  12.  In  the  second  square  of  the  third  row, 
it  will  be  found,  that  £  makes  f ;  and  that  f  make  £. 

25.  la  the  fourth  square  of  the  second  row,  it 
will  be  seen  that  1  half  is  f ;  and  in  the  second 
square  of  the  fourth  row,  i  is  f ;  both  together  make 
f ,  and  £  makes  £ . 

27.  In  the  second  square  of  the  fourth  row,  J  is 
the  same  as  f . 

33.  In  the  fifth  square  of  the  fourth  row,  it  will 
be  seen  that  £  (made  by  the  vertical  division)  con- 
tains ^y ;  and  in  the  fourth  square  of  the  fifth  row, 
•J  contains  ^j,  and  -|  contain  •/§  ;  and  in  the  second 
square  of  the  tenth  row,  y1^  contains  -£$. 

When  these  questions  are  performed  in  the  mind, 
the  pupil  will  explain  them  as  follows.  He  will 


SECT.  13.]  KEY.  165 

probably  do  it  without  assistance.  Twenty  twen- 
tieths make  one  whole  one.  £  of  20  is  5,  and  |-  of 
20  is  8,  and  ^  of  20  is  2  ;  therefore  £  is  ^,  f  is 
/^,  and  yiy  is  -£$.  All  the  examples  should  be  ex- 
plained in  the  same  manner. 

45.  In  the  8th  row,  the  7th  square  is  divided  ver- 
tically into  8  parts,  and  horizontally  into  7  parts  ; 
the  square,  therefore,  is  divided  into  56  parts  ;  3  of 
the  vertical  divisions,  or  f  ,  contain  f  £. 

51.  1  half  is  |,  and  £  is  f  ,  which  added  together 
make  f  . 

61.  f  is  ?8<y,  -£$  is-/u>  i  ls  -fv>  which  added  togeth- 
er make  J-f  . 

67.  f  is  T82-,  f  is  T92-,  which  added  together  make 
ij  ;  from  -ij  take  -f^,  and  there  remains  -ff-  ,  or  1. 

82.  It  will  be  easily  perceived  that  these  exam- 
ples do  not  differ  from  those  in  the  first  part  of  the 
section,  except  in  the  language  used.  They  must 
be  reduced  to  a  common  denominator,  and  then 
they  may  be  added  and  subtracted  as  easily  as 
whole  numbers,  f  is  |£,  and  f  is  T\,  and  both  to- 
gether make  J-f  ,  or  ly^. 
,  86.  £  is  f  ,  and  £  is  |.  If  f  be  taken  from  f  there 


B.  This  article  contains  only  a  practical  applica- 
tion of  the  preceding. 

3.  This  example  and  some  of  the  following  con- 
tain mixed  numbers,  but  they  are  quite  as  easy  as 
the  others.  The  whole  numbers  may  be  added 
separately,  and  the  fractions  reduced  to  a  common 
denominator,  and  then  added  as  in  other  cases,  and 
afterwards  joined  to  the  whole  numbers.  6  and  2 
are  8  ;  1  half  and  £  are  f  ,  making  in  the  whole  8f 
bushels. 

5.  6  and  2  are  8  ;  f  and  £  and  f  are  £J  or  1£J, 
which  joined  with  8  make  9£J. 


166  KEY.  [PART  ± 

C.  It  is  difficult  to  find  examples  which  will  aptly 
illustrate  this  operation.  It  can  be  done  more  con- 
veniently by  the  instructer.  Whenever  a  fraction 
occurs,  which  may  be  reduced  to  lower  terms,  if  it 
be  suggested  to  the  pupil,  he  will  readily  perceive 
it  and  do  it.  This  may  be  done  in  almost  any  part 
of  the  book,  but  more  especially  after  studying  the 
13th  section.  Perhaps  it  would  be  as  well  to  omit 
this  article  the  first  time  the  pupil  goes  through  the 
book,  and,  after  he  has  seen  the  use  of  the  operation, 
to  let  him  study  it.  It  may  be  illustrated  on  Plate 
III.  in  the  following  manner  : 

8.  £f .  Find  all  the  squares  which  are  divided 
into  24  parts.  There  are  4  squares  which  are  di- 
vided into  24  parts,  viz.  the;  8th  in  the  3d  row,  the 
3d  in  the  8th  row,  the  6th  in  the  4th  row,  and  the 
4th  in  the  6th  row.  Then  see  if  exactly  18  can  be 
found  in  one  or  more  of  the  vertical  divisions.  In 
the  6th  square  of  the  4th  row,  there  are  exactly  18 
divisions  in  three  vertical  divisions ;  but  those  3  ver- 
tical divisions  are  £  of  the  whole  square,  because  it 
is  divided  into  fourths  vertically ;  therefore  £f  are 
equal  to  £ . 

13.  £|  Find  the  squares  which  are  divided  into 
56  parts ;  they  are  the  8th  in  the  seventh  row,  and 
the  7th  in  the  8th  row ;  see  if  in  either  of  them, 
one  or  more  of  the  vertical  divisions  contain  exactly 
42  parts.  In  the  7th  of  the  8th  row,  6  vertical  di- 
visions contain  exactly  42  :  these  divisions  are  f  of 
the  square,  for  it  is  divided  vertically  into  8  parts. 
But  f  may  be  still  reduced  to  f ,  as  may  be  seen  by 
looking  on  the  3d  square  of  the  4th  row;  therefore 
f  |  is  equal  to  £. 


SECT.  14.]  KEY.  167 

SECTION  XIV. 

A.  THIS  section  contains  the  division  of  fractions 
by  whole  numbers,  and  the  multiplication  of  one  frac- 
tion by  another.  Though  these  operations  sometimes 
appear  to  be  division,  and  sometimes  multiplication, 
yet  there  is  actually  no  difference  in  the  operations. 

The  practical  examples  will  generally  show  how 
the  operations  are  to  be  performed,  but  it  will  be 
well  to  use  the  plate  for  young  pupils. 

1  and  2.  In  the  second  row,  the  2d  square  is  di- 
vided vertically  into  halves,  and  each  of  the  halves 
is  divided  into  halves  by  the  horizontal  line  ;  J  of  £ 
is  therefore  £  of  the  whole. 

3  and  4.  In  the  third  row,  the  2d  square  shows 
that  £  of  -J  is  £. 

16  and  17.  In  the  5th  row,  the  3d  square  shows 
that  £  of  £  is  T\  of  the  whole. 

33.  Since  f  of  a  share  signify  3  parts  of  a  share, 
it  is  evident  that  -J  of  the  three  parts  is  1  part,  that 
is,  i- 

39.  ^  signify  9  pieces  or  parts,  and  it  is  evident 
that  |  of  9  parts  is  3  parts,  that  is,  f . 

43.  We  cannot  take  -J  of  5  pieces ;  therefore  we 
must  take  £  of  -£,  which  is  ^,  and  f  is  5  times  as 
much  as  £ ;  therefore  ^  of  f  is  T5ff.  This  may  be 
readily  seen  on  the  plate.  In  the  sixth  row,  jthird 
square,  find  -|  by  the  vertical  division;  then,  these 
being  divided  each  into  three  parts  by  the  hori- 
zontal division,  and  -J  of  each  being  taken,  you  will 
have  y\. 

5*2.  In  the  4th  row,  the  3d  square  shows  that  -J  of 
£  is  TV,  and  §  must  be  twice  as  much,  or  fa. 

56.  In  the  fifth  row,  the  3d  square  shows  that  -J- 
of  |.  is  T2^  ;  but  |  must  be  twice  as  much  as  ^ ;  there- 
fore §  of  f  are  ^. 


168  KEY.  [PART 


78.  8f  is^,| 

79.  8|  is  Sf-,  |  of  |  is  ^,  consequently  J  of 
is  ££,  or  1«. 

86.  We  may  say,  £  of  8  f  is  2,  and  2|  over  ;  then 
2J.  is  32,  and  £  of  ^  is  £f  ;  hence  $  of  8$  is  2ff 
90.  |  of  18f  is  2§£,  and  £  is  3  times  as  much,  or 

7ft- 

B.  4.  It  would  take  1  man  4  times  9f  ,  or  37^ 
days  ;  and  7  men  would  do  it  in  ^  of  that  time,-  that 
is,  in  5^f  days. 


SECTION  XV. 

A.  THIS  section  contains  the  division  of  whole 
numbers  by  fractions,  and  fractions  by  fractions. 

1.  Since  there  are  §•  in  2,  it  is  evident  that  he 
could  give  them  to  6  boys  if  he  gave  them  £  apiece, 
but  if  he  gave  them  §  apiece,  he  could  give  them  to 
only  one  half  as  many,  or  3  boys. 

5.  If  ^  of  a  barrel  would  last  them  one  month,  it 
is  evident  that  4  barrels  would  last  20  months ;  but 
since  it  takes  f  of  a  barrel,  it  will  last  them  but  one 
half  as  long,  or  10  months. 

7.  6f  is  -2¥7-.  If  £  of  a  bushel  would  last  a  week, 
6f  bushels  would  last  27  weeks ;  but  since  it  takes 
J,  it  will  last  only  £  of  the  time,  or  9  weeks. 

13.  If  he  had  given  %  of  a  bushel  apiece,  he 
might  have  given  it  to  17  persons :  but  since  he  gave 
3  halves  apiece,  he  could  give  it  to  only  ^  of  that 
number,  that  is,  to  5  persons,  and  he  would  have  1 
bushel  left,  which  would  be  §  of  enough  for  another. 

23.  9f  is  £T<s  and  If  is  -y1--  If  it  had  been  only 
-f  of  a  dollar  a  barrel,  he  might  have  bought  66 
barrels  for  9f  dollars;  but  since  it  was  4^  a  bar- 


SECT.  15.]  KEY.  169 

rel,  he  could  buy  only  -^  of  that  number,  that  is,  6 
barrels. 

25  and  26.     Ans.  9f 

31.  4i  is  -2^,  and  9f  is  A£.  Now,  |  is  contained 
in  4^.  48  times,  and  -^  is  contained  only  ^  part  as 
many  times,  consequently  only  2^r  or  2f  . 


B.  1.  J  is  !%•;  consequently  5  pounds  can  be 
bought  for  %  of  a  dollar. 

3.  £•  is  T92-,  and  £  is  -f^.  If  he  had  given  only  T^. 
apiece,  he  could  have  given  it  to  9  persons;  but 
since  he  gave  -fg,  he  could  give  it  to  only  1  half  as 
many,  or  4£  persons. 

5.  |  is  ^  and  2.  is  ^.4.  If  a  pound  had  cost  ^ 
of  a  dollar,  14  pounds  could  be  bought  for  ££  of  a 
dollar  ;  but  since  it  costs  ^-,  only  ^  as  many  can  be 
bought  ;  that  is,  4f  pounds. 

9.  2.  is  £6.^  and  1-|  is  |^.  If  a  bushel  had  cost 
^y  of  a  dollar,  65  bushels  might  have  been  bought  ; 
but  since  it  cost  £g,  only  TJF  part  as  much  could  be 
bought  ;  that  is,  4T^  bushels. 

12.  f  is  ^F,  and  f  is  ^f  ;  T^  is  contained  in  j-%  15 
times,  but  -fa  is  contained  only  £  as  many  times  ; 
that  is,  3f  times. 

Miscellaneous  Examples. 

5.  f  of  a  penny  is  f  of  4  farthings.     Ans.  2f 
farthings. 

6.  f  of  12  pence.     Ans.  10  pence. 

7.  f  of  4  quarters  is  2  quarters  and  f  of  a  quar- 
ter ;  f  of  a  quarter  is  f  of  4  nails,  which  is  1|^  nails. 
Ans.  2  quarters,  1^  nails. 

13.  f  of  24  hours  is  15  hours. 

14.  |^  of  24  hours  is  14  hours  and  f  of  an  hour  ; 
f  of  60  minutes  is  24  minutes.     Ans.  14  hours,  24 
minutes. 

15 


170  KEY.  [PART  2. 

28.  There  being  4  farthings  in  a  penny,  1  farthing 
is  £  part  of  a  penny. 

30.  3  farthings  is  £  of  a  penny. 

31.  1  penny  is  -^  of  a  shilling,  because  there  are 
12  pence  in  a  shilling. 

34.  5  pence  is  -j^-  of  a  shilling. 
41.  1  shilling  is  -^  of  a  pound. 
43.  3  shillings  is  ^  of  a  pound. 

48.  1  farthing  is  ^  of  one  shilling 

49.  2  farthings  is  ^2F,  or  ?V  of  a  shilling.     5  far- 
things is  ¥%-  of  a  shilling. 

51.  1  penny  is  ^fa  of  1  pound.     7  pence  is  ^J^ 

Ofcfl. 

59.  3s.  5d.  is  41  pence,  which  is  £fc  of  £1. 

75.  1  nail  is  ^  of  a  yard.    5  nails  is  -fa  of  a  yard. 

89.  1  oz.  is  -j^.of  1  Ib.     15  oz.  is  -}-£  of  1  Ib. 

91.  1  Ib.  is  ^  of  1  quarter.  9  Ibs.  is  ^  of  1 
quarter. 

100.  At  the  end  of  1  hour,  they  would  be  7  and  £ 
miles  apart ;  in  7  hours,  7  times  7f ,  which  is  54g 
miles. 

121.  This  is  the  principle  of  fellowship ;  3  shil- 
lings were  paid ;  one  paid  £,  the  other  f . 

122.  One  paid  f ,  the  other  f . 

123.  20  dollars  were  paid  in  the  whole ;  one  paid 
^5-,  another  ^  and  the  third  •£>. 

126.  3  and  4  and  5  are  12.  The  first  put  in  T^ ; 
the  second  -£% ;  the  third  T5^. 

129.  4  dollars  for  2  months  is  the  same  as  8  dol- 
lars for  1  month  ;  3  dollars  for  3  months  is  the  same 
as  9  dollars  for  1  month ;  and  2  dollars  for  4  months 
is  the  same  as  8  dollars  for  1  month.  The  question 
is  the  same  as  if  A  had  put  in  8  dollars,  B  9  dollars, 
and  C  8  dollars.  A  must  have  ^,  B  ^\,  and  C 
^,  of  100  dollars. 

131.  A's  money  was  in  4  times  as  long  as  C's. 
It  is  the  same  as  if  A  had  put  in  8  dollars  for  the 


PART  2.]  KEY.  171 

same  time,  and  B  8  dollars  for  the  same  time.      A 
must  have  ^  B  4-,  and  C  ^,  of  88  dollars. 

The  examples  "127,  128,  129,  130,  and  131,  are 
double  or  compound  fellowship. 

139.  The  interest  of  50  dollars  for  1  year  and  6 
months  is  4  dollars  and  50  cents,  and  for  1  month  it 
is  25  cents.     The  interest  of  7  dollars  for  18  months 
(a  dollar  is  £  of  a  cent  a  month)  is  63  cents.     The 
whole  amounts  to  5  dollars  and  38  cents. 

140.  The  interest  of  200  dollars  for  1|  year  is 
16  dollars.     The  interest  of  67  dollars  is  67  cents 
for  every  2  months  ;  for  16  months  it  will  be  8  times 
67  cents,  which  are  5  dollars  and  36  cents.     The 
whole  interest  is  21  dollars  and  36  cents. 

143.  The  interest  of  100  dollars  for  2£  years, 
is  13  dollars  and  50  cents.  The  interest  of  100 
dollars  for  60  days  would  be  1  dollar ;  the  interest 
for  20  days  will  be  %  of  a  dollar,  or  33£  cents. 
The  interest  of  1  dollar  for  2£  years  is  13£  cents ; 
for  10  dollars  the  interest  would  be  1  dollar  and 
35  cents,  and  for  30  dollars,  4  dollars  and  5  cents. 
The  interest  of  7  dollars  for  2£  years  is  7  times 
13^  cents,  or  94  j-  cents.  The  interest  of  37  dollars 
for  60  days  would  be  37  cents,  and  for  20  days  £  of 
37  cents,  or  12£  cents.  The  whole  interest  is  18 
dollars  and  95£  cents. 

146.  They  would  both  together  do  f  of  the  work 
in  1  day,  and  it  would  take  them  £  of  a  day  to  do 
the  other  £.     Ans.  l£  day. 

147.  -|  would  be  done  in  1  day,  and  it  would  take 
£  of  a  day  to  do  the  other  £.     Ans.  l-i  day. 

H(9-  They  both  together  consume  £  of  a  bushel 
hi  a  week,  but  the  woman  alone  consumes  only  £  of 
a  bushel  in  a  week.  That  is,  they  both  together 
consume  -f$  in  a  week,  but  the  woman  alone  only 
^ ;  consequently,  the  man  alone  would  consume  -fa ; 
and  a  bushel  would  last  him  3|  weeks. 


173  KEY.  [PARTS. 

152.  A  and  B  can  build  ^  of  it  in  1  day ;  A,  B, 
and  C,  can  build  |  of  it  in  1  day ;  the  difference  be- 
tween £  and  £  is  ft ;  therefore  C  can  build  ^  of 
it  in  1  day ;  and  it  would  take  him  13£  days  to 
build  it  alone. 

164.  Find  how  much  they  might  eat  in  a  day,  in 
order  to  make  it  last  1  month,  and  then  it  will  be 
easy  to  find  how  much  they  may  eat  in  a  day,  to 
make  it  last  11  months. 

167.  The  money  is  7  parts  of  the  whole,  and  the 
purse  one  part ;  consequently  the  money  is  £,  and 
the  purse  £,  of  16. 

170.  He  gave  one  part  for  the  apple,  2  parts  for 
the  orange,  and  4  parts  for  the  melon.  These  make 
7  parts.  The  apple  3  cents,  the  orange  6  cents,  and 
the  melon  12  cents. 

175.  If  to  a  number  half  of  itself  be  added,  the 
sum  is  f  of  that  number ;  hence  subtract  2£  from 
100,  and  the  remainder  is  f  of  the  number  of  geese 
that  he  had. 

180.  This  must  be  reduced  to  6ths.     1  half  is  g, 
and  i  is  §,  and  the  number  itself  is  £ .     If,  therefore, 
to  the  whole  number  its  half  and  its  third  be  added, 
the  sum  will  be  Jgi ;  hence,  77  is  -y-  of  the  number. 

181.  £  is  f ;  therefore  if  to  a  number  £  and  £  of 
itself  be  added,  the  whole  number  will  be  J;  but 
when  18  more  is  added  to  J,  the  first  number  is 
doubled ;  that  is,  the  number  is  f  of  the  first  num- 
ber ;  therefore  18  is  £  of  the  number. 


VA  04342 


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